A Visual Guide to How Diffusion Models Work

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A Visual Guide to How Diffusion Models Work

This article is aimed at those who want to understand exactly how Diffusion Models work, with no prior knowledge expected. I’ve tried to use illustrations wherever possible to provide visual intuitions on each part of these models. I’ve kept mathematical notation and equations to a minimum, and where they are necessary I’ve tried to define […]

Intro

I’ve framed this article around three main questions:

What exactly is it that diffusion models learn?
How and why do diffusion models work?
Once you’ve trained a model, how do you get useful stuff out of it?

The examples will be based on the glyffuser, a minimal text-to-image diffusion model that I previously implemented and wrote about. The architecture of this model is a standard text-to-image denoising diffusion model without any bells or whistles. It was trained to generate pictures of new “Chinese” glyphs from English definitions. Have a look at the picture below — even if you’re not familiar with Chinese writing, I hope you’ll agree that the generated glyphs look pretty similar to the real ones!

Random examples of glyffuser training data (left) and generated data (right).

What exactly is it that diffusion models learn?

Generative Ai models are often said to take a big pile of data and “learn” it. For text-to-image diffusion models, the data takes the form of pairs of images and descriptive text. But what exactly is it that we want the model to learn? First, let’s forget about the text for a moment and concentrate on what we are trying to generate: the images.

Probability distributions

Broadly, we can say that we want a generative AI model to learn the underlying probability distribution of the data. What does this mean? Consider the one-dimensional normal (Gaussian) distribution below, commonly written 𝒩(μ,σ²) and parameterized with mean μ = 0 and variance σ² = 1. The black curve below shows the probability density function. We can sample from it: drawing values such that over a large number of samples, the set of values reflects the underlying distribution. These days, we can simply write something like x = random.gauss(0, 1) in Python to sample from the standard normal distribution, although the computational sampling process itself is non-trivial!

Values sampled from an underlying distribution (here, the standard normal 𝒩(0,1)) can then be used to estimate the parameters of that distribution.

We could think of a set of numbers sampled from the above normal distribution as a simple dataset, like that shown as the orange histogram above. In this particular case, we can calculate the parameters of the underlying distribution using maximum likelihood estimation, i.e. by working out the mean and variance. The normal distribution estimated from the samples is shown by the dotted line above. To take some liberties with terminology, you might consider this as a simple example of “learning” an underlying probability distribution. We can also say that here we explicitly learnt the distribution, in contrast with the implicit methods that diffusion models use.

Conceptually, this is all that generative AI is doing — learning a distribution, then sampling from that distribution!

Data representations

What, then, does the underlying probability distribution of a more complex dataset look like, such as that of the image dataset we want to use to train our diffusion model?

First, we need to know what the representation of the data is. Generally, a machine learning (ML) model requires data inputs with a consistent representation, i.e. format. For the example above, it was simply numbers (scalars). For images, this representation is commonly a fixed-length vector.

The image dataset used for the glyffuser model is ~21,000 pictures of Chinese glyphs. The images are all the same size, 128 × 128 = 16384 pixels, and greyscale (single-channel color). Thus an obvious choice for the representation is a vector x of length 16384, where each element corresponds to the color of one pixel: x = (x₁,x₂,…,x₁₆₃₈₄). We can call the domain of all possible images for our dataset “pixel space”.

An example glyph with pixel values labelled (downsampled to 32 × 32 pixels for readability).

Dataset visualization

We make the assumption that our individual data samples, x, are actually sampled from an underlying probability distribution, q(x), in pixel space, much as the samples from our first example were sampled from an underlying normal distribution in 1-dimensional space. Note: the notation x ∼ q(x) is commonly used to mean: “the random variable x sampled from the probability distribution q(x).”

This distribution is clearly much more complex than a Gaussian and cannot be easily parameterized — we need to learn it with a ML model, which we’ll discuss later. First, let’s try to visualize the distribution to gain a better intution.

As humans find it difficult to see in more than 3 dimensions, we need to reduce the dimensionality of our data. A small digression on why this works: the manifold hypothesis posits that natural datasets lie on lower dimensional manifolds embedded in a higher dimensional space — think of a line embedded in a 2-D plane, or a plane embedded in 3-D space. We can use a dimensionality reduction technique such as UMAP to project our dataset from 16384 to 2 dimensions. The 2-D projection retains a lot of structure, consistent with the idea that our data lie on a lower dimensional manifold embedded in pixel space. In our UMAP, we see two large clusters corresponding to characters in which the components are arranged either horizontally (e.g. 明) or vertically (e.g. 草). An interactive version of the plot below with popups on each datapoint is linked here.

Click here for an interactive version of this plot.

Let’s now use this low-dimensional UMAP dataset as a visual shorthand for our high-dimensional dataset. Remember, we assume that these individual points have been sampled from a continuous underlying probability distribution q(x). To get a sense of what this distribution might look like, we can apply a KDE (kernel density estimation) over the UMAP dataset. (Note: this is just an approximation for visualization purposes.)

This gives a sense of what q(x) should look like: clusters of glyphs correspond to high-probability regions of the distribution. The true q(x) lies in 16384 dimensions — this is the distribution we want to learn with our diffusion model.

We showed that for a simple distribution such as the 1-D Gaussian, we could calculate the parameters (mean and variance) from our data. However, for complex distributions such as images, we need to call on ML methods. Moreover, what we will find is that for diffusion models in practice, rather than parameterizing the distribution directly, they learn it implicitly through the process of learning how to transform noise into data over many steps.

Takeaway

The aim of generative AI such as diffusion models is to learn the complex probability distributions underlying their training data and then sample from these distributions.

How and why do diffusion models work?

Diffusion models have recently come into the spotlight as a particularly effective method for learning these probability distributions. They generate convincing images by starting from pure noise and gradually refining it. To whet your interest, have a look at the animation below that shows the denoising process generating 16 samples.

In this section we’ll only talk about the mechanics of how these models work but if you’re interested in how they arose from the broader context of generative models, have a look at the further reading section below.

What is “noise”?

Let’s first precisely define noise, since the term is thrown around a lot in the context of diffusion. In particular, we are talking about Gaussian noise: consider the samples we talked about in the section about probability distributions. You could think of each sample as an image of a single pixel of noise. An image that is “pure Gaussian noise”, then, is one in which each pixel value is sampled from an independent standard Gaussian distribution, 𝒩(0,1). For a pure noise image in the domain of our glyph dataset, this would be noise drawn from 16384 separate Gaussian distributions. You can see this in the previous animation. One thing to keep in mind is that we can choose the means of these noise distributions, i.e. center them, on specific values — the pixel values of an image, for instance.

For convenience, you’ll often find the noise distributions for image datasets written as a single multivariate distribution 𝒩(0,I) where I is the identity matrix, a covariance matrix with all diagonal entries equal to 1 and zeroes elsewhere. This is simply a compact notation for a set of multiple independent Gaussians — i.e. there are no correlations between the noise on different pixels. In the basic implementations of diffusion models, only uncorrelated (a.k.a. “isotropic”) noise is used. This article contains an excellent interactive introduction on multivariate Gaussians.

Diffusion process overview

Below is an adaptation of the somewhat-famous diagram from Ho et al.’s seminal paper “Denoising Diffusion Probabilistic Models” which gives an overview of the whole diffusion process:

Diagram of the diffusion process adapted from Ho *et al*. 2020. The glyph 锂, meaning “lithium”, is used as a representative sample from the dataset.

I found that there was a lot to unpack in this diagram and simply understanding what each component meant was very helpful, so let’s go through it and define everything step by step.

We previously used x ∼ q(x) to refer to our data. Here, we’ve added a subscript, xₜ, to denote timestep t indicating how many steps of “noising” have taken place. We refer to the samples noised a given timestep as x ∼ q(xₜ). x₀ is clean data and xₜ (t = T) ∼ 𝒩(0,1) is pure noise.

We define a forward diffusion process whereby we corrupt samples with noise. This process is described by the distribution q(xₜ|xₜ₋₁). If we could access the hypothetical reverse process q(xₜ₋₁|xₜ), we could generate samples from noise. As we cannot access it directly because we would need to know x₀, we use ML to learn the parameters, θ, of a model of this process, 𝑝θ(𝑥ₜ₋₁∣𝑥ₜ). (That should be p subscript θ but medium cannot render it.)

In the following sections we go into detail on how the forward and reverse diffusion processes work.

Forward diffusion, or “noising”

Used as a verb, “noising” an image refers to applying a transformation that moves it towards pure noise by scaling down its pixel values toward 0 while adding proportional Gaussian noise. Mathematically, this transformation is a multivariate Gaussian distribution centered on the pixel values of the preceding image.

In the forward diffusion process, this noising distribution is written as q(xₜ|xₜ₋₁) where the vertical bar symbol “|” is read as “given” or “conditional on”, to indicate the pixel means are passed forward from q(xₜ₋₁) At t = T where T is a large number (commonly 1000) we aim to end up with images of pure noise (which, somewhat confusingly, is also a Gaussian distribution, as discussed previously).

The marginal distributions q(xₜ) represent the distributions that have accumulated the effects of all the previous noising steps (marginalization refers to integration over all possible conditions, which recovers the unconditioned distribution).

Since the conditional distributions are Gaussian, what about their variances? They are determined by a variance schedule that maps timesteps to variance values. Initially, an empirically determined schedule of linearly increasing values from 0.0001 to 0.02 over 1000 steps was presented in Ho et al. Later research by Nichol & Dhariwal suggested an improved cosine schedule. They state that a schedule is most effective when the rate of information destruction through noising is relatively even per step throughout the whole noising process.

Forward diffusion intuition

As we encounter Gaussian distributions both as pure noise q(xₜ, t = T) and as the noising distribution q(xₜ|xₜ₋₁), I’ll try to draw the distinction by giving a visual intuition of the distribution for a single noising step, q(x₁∣x₀), for some arbitrary, structured 2-dimensional data:

Each noising step q(xₜ|xₜ₋₁) is a Gaussian distribution conditioned on the previous step.

The distribution q(x₁∣x₀) is Gaussian, centered around each point in x₀, shown in blue. Several example points x₀⁽ⁱ⁾ are picked to illustrate this, with q(x₁∣x₀ = x₀⁽ⁱ⁾) shown in orange.

In practice, the main usage of these distributions is to generate specific instances of noised samples for training (discussed further below). We can calculate the parameters of the noising distributions at any timestep t directly from the variance schedule, as the chain of Gaussians is itself also Gaussian. This is very convenient, as we don’t need to perform noising sequentially—for any given starting data x₀⁽ⁱ⁾, we can calculate the noised sample xₜ⁽ⁱ⁾ by sampling from q(xₜ∣x₀ = x₀⁽ⁱ⁾) directly.

Forward diffusion visualization

Let’s now return to our glyph dataset (once again using the UMAP visualization as a visual shorthand). The top row of the figure below shows our dataset sampled from distributions noised to various timesteps: xₜ ∼ q(xₜ). As we increase the number of noising steps, you can see that the dataset begins to resemble pure Gaussian noise. The bottom row visualizes the underlying probability distribution q(xₜ).

The dataset xₜ (above) sampled from its probability distribution q(xₜ) (below) at different noising timesteps.

Reverse diffusion overview

It follows that if we knew the reverse distributions q(xₜ₋₁∣xₜ), we could repeatedly subtract a small amount of noise, starting from a pure noise sample xₜ at t = T to arrive at a data sample x₀ ∼ q(x₀). In practice, however, we cannot access these distributions without knowing x₀ beforehand. Intuitively, it’s easy to make a known image much noisier, but given a very noisy image, it’s much harder to guess what the original image was.

So what are we to do? Since we have a large amount of data, we can train an ML model to accurately guess the original image that any given noisy image came from. Specifically, we learn the parameters θ of an ML model that approximates the reverse noising distributions, pθ(xₜ₋₁ ∣ xₜ) for t = 0, …, T. In practice, this is embodied in a single noise prediction model trained over many different samples and timesteps. This allows it to denoise any given input, as shown in the figure below.

The ML model predicts added noise at any given timestep t.

Next, let’s go over how this noise prediction model is implemented and trained in practice.

How the model is implemented

First, we define the ML model — generally a deep neural network of some sort — that will act as our noise prediction model. This is what does the heavy lifting! In practice, any ML model that inputs and outputs data of the correct size can be used; the U-net, an architecture particularly suited to learning images, is what we use here and frequently chosen in practice. More recent models also use vision transformers.

We use the U-net architecture (Ronneberger *et al*. 2015) for our ML noise prediction model. We train the model by minimizing the difference between predicted and actual noise.

Then we run the training loop depicted in the figure above:

We take a random image from our dataset and noise it to a random timestep tt. (In practice, we speed things up by doing many examples in parallel!)
We feed the noised image into the ML model and train it to predict the (known to us) noise in the image. We also perform timestep conditioning by feeding the model a timestep embedding, a high-dimensional unique representation of the timestep, so that the model can distinguish between timesteps. This can be a vector the same size as our image directly added to the input (see here for a discussion of how this is implemented).
The model “learns” by minimizing the value of a loss function, some measure of the difference between the predicted and actual noise. The mean square error (the mean of the squares of the pixel-wise difference between the predicted and actual noise) is used in our case.
Repeat until the model is well trained.

Note: A neural network is essentially a function with a huge number of parameters (on the order of 10⁶ for the glyffuser). Neural network ML models are trained by iteratively updating their parameters using backpropagation to minimize a given loss function over many training data examples. This is an excellent introduction. These parameters effectively store the network’s “knowledge”.

A noise prediction model trained in this way eventually sees many different combinations of timesteps and data examples. The glyffuser, for example, was trained over 100 epochs (runs through the whole data set), so it saw around 2 million data samples. Through this process, the model implicity learns the reverse diffusion distributions over the entire dataset at all different timesteps. This allows the model to sample the underlying distribution q(x₀) by stepwise denoising starting from pure noise. Put another way, given an image noised to any given level, the model can predict how to reduce the noise based on its guess of what the original image. By doing this repeatedly, updating its guess of the original image each time, the model can transform any noise to a sample that lies in a high-probability region of the underlying data distribution.

Reverse diffusion in practice

We can now revisit this video of the glyffuser denoising process. Recall a large number of steps from sample to noise e.g. T = 1000 is used during training to make the noise-to-sample trajectory very easy for the model to learn, as changes between steps will be small. Does that mean we need to run 1000 denoising steps every time we want to generate a sample?

Luckily, this is not the case. Essentially, we can run the single-step noise prediction but then rescale it to any given step, although it might not be very good if the gap is too large! This allows us to approximate the full sampling trajectory with fewer steps. The video above uses 120 steps, for instance (most implementations will allow the user to set the number of sampling steps).

Recall that predicting the noise at a given step is equivalent to predicting the original image x₀, and that we can access the equation for any noised image deterministically using only the variance schedule and x₀. Thus, we can calculate xₜ₋ₖ based on any denoising step. The closer the steps are, the better the approximation will be.

Too few steps, however, and the results become worse as the steps become too large for the model to effectively approximate the denoising trajectory. If we only use 5 sampling steps, for example, the sampled characters don’t look very convincing at all:

There is then a whole literature on more advanced sampling methods beyond what we’ve discussed so far, allowing effective sampling with much fewer steps. These often reframe the sampling as a differential equation to be solved deterministically, giving an eerie quality to the sampling videos — I’ve included one at the end if you’re interested. In production-level models, these are usually preferred over the simple method discussed here, but the basic principle of deducing the noise-to-sample trajectory is the same. A full discussion is beyond the scope of this article but see e.g. this paper and its corresponding implementation in the Hugging Face diffusers library for more information.

Alternative intuition from score function

To me, it was still not 100% clear why training the model on noise prediction generalises so well. I found that an alternative interpretation of diffusion models known as “score-based modeling” filled some of the gaps in intuition (for more information, refer to Yang Song’s definitive article on the topic.)

The dataset xₜ sampled from its probability distribution q(xₜ) at different noising timesteps; below, we add the score function ∇ₓ log q(xₜ).

I try to give a visual intuition in the bottom row of the figure above: essentially, learning the noise in our diffusion model is equivalent (to a constant factor) to learning the score function, which is the gradient of the log of the probability distribution: ∇ₓ log q(x). As a gradient, the score function represents a vector field with vectors pointing towards the regions of highest probability density. Subtracting the noise at each step is then equivalent to moving following the directions in this vector field towards regions of high probability density.

As long as there is some signal, the score function effectively guides sampling, but in regions of low probability it tends towards zero as there is little to no gradient to follow. Using many steps to cover different noise levels allows us to avoid this, as we smear out the gradient field at high noise levels, allowing sampling to converge even if we start from low probability density regions of the distribution. The figure shows that as the noise level is increased, more of the domain is covered by the score function vector field.

Summary

The aim of diffusion models is learn the underlying probability distribution of a dataset and then be able to sample from it. This requires forward and reverse diffusion (noising) processes.
The forward noising process takes samples from our dataset and gradually adds Gaussian noise (pushes them off the data manifold). This forward process is computationally efficient because any level of noise can be added in closed form a single step.
The reverse noising process is challenging because we need to predict how to remove the noise at each step without knowing the original data point in advance. We train a ML model to do this by giving it many examples of data noised at different timesteps.
Using very small steps in the forward noising process makes it easier for the model to learn to reverse these steps, as the changes are small.
By applying the reverse noising process iteratively, the model refines noisy samples step by step, eventually producing a realistic data point (one that lies on the data manifold).

Takeaway

Diffusion models are a powerful framework for learning complex data distributions. The distributions are learnt implicitly by modelling a sequential denoising process. This process can then be used to generate samples similar to those in the training distribution.

Once you’ve trained a model, how do you get useful stuff out of it?

Earlier uses of generative AI such as “This Person Does Not Exist” (ca. 2019) made waves simply because it was the first time most people had seen AI-generated photorealistic human faces. A generative adversarial network or “GAN” was used in that case, but the principle remains the same: the model implicitly learnt a underlying data distribution — in that case, human faces — then sampled from it. So far, our glyffuser model does a similar thing: it samples randomly from the distribution of Chinese glyphs.

The question then arises: can we do something more useful than just sample randomly? You’ve likely already encountered text-to-image models such as Dall-E. They are able to incorporate extra meaning from text prompts into the diffusion process — this in known as conditioning. Likewise, diffusion models for scientific scientific applications like protein (e.g. Chroma, RFdiffusion, AlphaFold3) or inorganic crystal structure generation (e.g. MatterGen) become much more useful if can be conditioned to generate samples with desirable properties such as a specific symmetry, bulk modulus, or band gap.

Conditional distributions

We can consider conditioning as a way to guide the diffusion sampling process towards particular regions of our probability distribution. We mentioned conditional distributions in the context of forward diffusion. Below we show how conditioning can be thought of as reshaping a base distribution.

A simple example of a joint probability distribution p(x, y), shown as a contour map, along with its two marginal 1-D probability distributions, p(x) and p(y). The highest points of p(x, y) are at (x₁, y₁) and (x₂, y₂). The conditional distributions p(x∣y = y₁) and p(x∣y = y₂) are shown overlaid on the main plot.

Consider the figure above. Think of p(x) as a distribution we want to sample from (i.e., the images) and p(y) as conditioning information (i.e., the text dataset). These are the marginal distributions of a joint distribution p(x, y). Integrating p(x, y) over y recovers p(x), and vice versa.

Sampling from p(x), we are equally likely to get x₁ or x₂. However, we can condition on p(y = y₁) to obtain p(x∣y = y₁). You can think of this as taking a slice through p(x, y) at a given value of y. In this conditioned distribution, we are much more likely to sample at x₁ than x₂.

In practice, in order to condition on a text dataset, we need to convert the text into a numerical form. We can do this using large language model (LLM) embeddings that can be injected into the noise prediction model during training.

Embedding text with an LLM

In the glyffuser, our conditioning information is in the form of English text definitions. We have two requirements: 1) ML models prefer fixed-length vectors as input. 2) The numerical representation of our text must understand context — if we have the words “lithium” and “element” nearby, the meaning of “element” should be understood as “chemical element” rather than “heating element”. Both of these requirements can be met by using a pre-trained LLM.

The diagram below shows how an LLM converts text into fixed-length vectors. The text is first tokenized (LLMs break text into tokens, small chunks of characters, as their basic unit of interaction). Each token is converted into a base embedding, which is a fixed-length vector of the size of the LLM input. These vectors are then passed through the pre-trained LLM (here we use the encoder portion of Google’s T5 model), where they are imbued with additional contextual meaning. We end up with a array of n vectors of the same length d, i.e. a (n, d) sized tensor.

We can convert text to a numerical embedding imbued with contextual meaning using a pre-trained LLM.

Note: in some models, notably Dall-E, additional image-text alignment is performed using contrastive pretraining. Imagen seems to show that we can get away without doing this.

Training the diffusion model with text conditioning

The exact method that this embedding vector is injected into the model can vary. In Google’s Imagen model, for example, the embedding tensor is pooled (combined into a single vector in the embedding dimension) and added into the data as it passes through the noise prediction model; it is also included in a different way using cross-attention (a method of learning contextual information between sequences of tokens, most famously used in the transformer models that form the basis of LLMs like ChatGPT).

Conditioning information can be added *via* multiple different methods but the training loss remains the same.

In the glyffuser, we only use cross-attention to introduce this conditioning information. While a significant architectural change is required to introduce this additional information into the model, the loss function for our noise prediction model remains exactly the same.

Testing the conditioned diffusion model

Let’s do a simple test of the fully trained conditioned diffusion model. In the figure below, we try to denoise in a single step with the text prompt “Gold”. As touched upon in our interactive UMAP, Chinese characters often contain components known as radicals which can convey sound (phonetic radicals) or meaning (semantic radicals). A common semantic radical is derived from the character meaning “gold”, “金”, and is used in characters that are in some broad sense associated with gold or metals.

Even with a single sampling step, conditioning guides denoising towards the relevant regions of the probability distribution.

The figure shows that even though a single step is insufficient to approximate the denoising trajectory very well, we have moved into a region of our probability distribution with the “金” radical. This indicates that the text prompt is effectively guiding our sampling towards a region of the glyph probability distribution related to the meaning of the prompt. The animation below shows a 120 step denoising sequence for the same prompt, “Gold”. You can see that every generated glyph has either the 釒 or 钅 radical (the same radical in traditional and simplified Chinese, respectively).

Takeaway

Conditioning enables us to sample meaningful outputs from diffusion models.

Further remarks

I found that with the help of tutorials and existing libraries, it was possible to implement a working diffusion model despite not having a full understanding of what was going on under the hood. I think this is a good way to start learning and highly recommend Hugging Face’s tutorial on training a simple diffusion model using their diffusers Python library (which now includes my small bugfix!).

I’ve omitted some topics that are crucial to how production-grade diffusion models function, but are unnecessary for core understanding. One is the question of how to generate high resolution images. In our example, we did everything in pixel space, but this becomes very computationally expensive for large images. The general approach is to perform diffusion in a smaller space, then upscale it in a separate step. Methods include latent diffusion (used in Stable Diffusion) and cascaded super-resolution models (used in Imagen). Another topic is classifier-free guidance, a very elegant method for boosting the conditioning effect to give much better prompt adherence. I show the implementation in my previous post on the glyffuser and highly recommend this article if you want to learn more.

Fun extras

Diffusion sampling using the DPMSolverSDEScheduler developed by Katherine Crowson and implemented in Hugging Face diffusers—note the smooth transition from noise to data.

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Fluor Corp. and Axens Group have been awarded key contracts for America First Refining’s (AFR) proposed grassroots refinery at the Port of Brownsville, Tex., advancing development of what would be the first new US refinery to be built in more than 50 years. Fluor will execute front-end engineering and design (FEED) for the project, while Axens will serve as technology licensor of core refining process technologies to be used at the site, the service providers said in separate Apr. 7 releases. The AFR refinery is designed to process more than 60 million bbl/year—or about 164,400 b/d—of US light shale crude into transportation fuels, including gasoline, diesel, and jet fuel. Contract details Without disclosing a specific value of its contract, Fluor said the scope of its FEED study will cover early-stage engineering and design required to define project execution, cost, and schedule based on a complex that will incorporate commercially proven technologies to improve efficiency and emissions performance while processing domestic shale crude. As technology licensor, Axens said it will deliver process technologies for key refining units at the site, including those for: Naphtha, diesel hydrotreating. Continuous catalytic reforming. Isomerization. Alongside supporting improved fuel-quality specifications, the unspecified technologies to be supplied for the refinery will also help to reduce overall energy consumption at the site. Axens—which confirmed its involvement since 2017 in working with AFR on early-stage development of the project—said this latest licensing agreement will also cover engineering support, equipment, catalysts, and services across the refinery’s process configuration. Project background, commercial framework Upon first announcing the project in March 2026, AFR said the proposed development came alongside an already signed 20-year offtake agreement with a global integrated oil company covering 1.2 billion bbl of US light shale crude, as well as capital investment to support construction. As part of the

EIA: US crude inventories up 3.1 million bbl

US crude oil inventories for the week ended Apr. 3, excluding the Strategic Petroleum Reserve, increased by 3.1 million bbl from the previous week, according to data from the US Energy Information Administration (EIA). At 464.7 million bbl, US crude oil inventories are about 2% above the 5-year average for this time of year, the EIA report indicated. EIA said total motor gasoline inventories decreased by 1.6 million bbl from last week and are about 3% above the 5-year average for this time of year. Finished gasoline inventories increased while blending components inventories decreased last week. Distillate fuel inventories decreased by 3.1 million bbl last week and are about 5% below the 5-year average for this time of year. Propane-propylene inventories increased by 600,000 bbl from last week and are 71% above the 5-year average for this time of year, EIA said. US crude oil refinery inputs averaged 16.3 million b/d for the week ended Apr. 3, which was 129,000 b/d less than the previous week’s average. Refineries operated at 92% of capacity. Gasoline production decreased, averaging 9.4 million b/d. Distillate fuel production increased, averaging 5.0 million b/d. US crude oil imports averaged 6.3 million b/d, down 130,000 b/d from the previous week. Over the last 4 weeks, crude oil imports averaged about 6.6 million b/d, 9.1% more than the same 4-week period last year. Total motor gasoline imports averaged 571,000 b/d. Distillate fuel imports averaged 152,000 b/d.

Oil prices plunge as Iran war tensions ease amid tentative Hormuz reopening

Crude oil prices plunged sharply on Apr. 7 after US President Donald Trump announced a conditional 2-week ceasefire agreement with Iran, contingent on reopening the Strait of Hormuz and restoring safe passage for energy shipments. Both Brent and WTI crude oil fell towards $95/bbl, marking their largest single-day decline since 2020. Under the agreement, Iran signaled willingness to halt attacks on shipping and allow transit through Hormuz while broader negotiations continue. The US also indicated it would assist in clearing a backlog of tankers and stabilizing maritime traffic. Benchmark crude prices initially surged above $110/bbl in early April amid fears of prolonged supply disruption after Iran effectively restricted traffic through the strait—a corridor responsible for roughly 20% of global oil flows. The blockade, triggered by escalating US-Iran hostilities, caused tanker traffic to collapse and stranded millions of barrels of crude and refined products in the region. Despite the price correction, analysts caution that supply disruptions and infrastructure damage will continue to constrain markets. The conflict has already impaired regional energy assets, including LNG infrastructure in Qatar, and forced producers across the Middle East to curtail output or delay exports. The US Energy Information Administration (EIA) warned that fuel prices may remain elevated for months even if flows normalize, citing logistical bottlenecks, depleted inventories, and continued geopolitical uncertainty. “In theory, the 10–13 million b/d of crude oil and product supply stranded behind the Strait should now be gradually released. Whether the pre-March status quo will be re-established depends entirely on whether the truce can be turned into a permanent peace during the negotiations in Pakistan,” said Tamas Varga, analyst, PVM Oil Associates. “What appears evident, at least for now, is that the current quarter, the April–June period, will be the tightest, as the scarcity of available oil, both crude and refined

EIA: Brent crude to reach $115/bbl in second-quarter 2026

Global oil markets have entered a period of acute volatility, with prices expected to surge into second-quarter 2026 as war-driven supply disruptions in the Middle East constrain flows through the Strait of Hormuz, according to the US Energy Information Administration (EIA)’s April Short-Term Energy Outlook. The agency estimates that Brent crude averaged $103/bbl in March and will climb further to a quarterly peak of about $115/bbl in second-quarter 2026, reflecting a sharp tightening in global supply following widespread production shut-ins across key Gulf producers. The disruption stems from the effective closure of the Strait of Hormuz, a critical chokepoint that typically carries nearly 20% of global oil supply. The US-Iran war in the region has forced producers including Saudi Arabia, Iraq, Kuwait, and the UAE to curtail output significantly. EIA estimates that crude production shut-ins averaged 7.5 million b/d in March and will rise to a peak of 9.1 million b/d in April. In this outlook, EIA assumes the conflict does not persist past April and that traffic through the Strait of Hormuz gradually resumes. Under those assumptions, EIA expects production shut-ins will fall to 6.7 million b/d in May and return close to pre-conflict levels in late 2026. The scale of the outage has rapidly flipped the market from prior expectations of oversupply into a pronounced deficit, with global inventories drawing sharply during the second quarter. Despite an assumption that the conflict does not persist beyond April, the agency warns that supply chains will take months to normalize, keeping a geopolitical risk premium embedded in prices through late 2026. EIA forecasts the Brent crude oil price will fall below $90/bbl in fourth-quarter 2026 and average $76/bbl in 2027, about $23/bbl higher than in its February STEO forecast. This price forecast is highly dependent on EIA’s assumptions of both the

Hillwood, PowerHouse Advance $20B Joliet Data Campus as Midwest AI Buildout Accelerates

The approval of the Joliet Technology Center signals that the Chicago region is being pulled into the Midwest’s next phase of AI infrastructure development, one that has so far been led by Ohio and defined by scale, power demand, and rising public scrutiny. It also underscores a growing reality: local governments are beginning to understand exactly what that shift entails. On March 19, 2026, the Joliet City Council voted 8–1 to approve the conditional annexation of roughly 795 acres for the proposed Joliet Technology Center, a $20 billion data center campus backed by Hillwood and PowerHouse Data Centers. The site, near Rowell and Bernhard Roads on Joliet’s east side, is planned as a 24-building, multi-phase development that would rank among the most consequential digital infrastructure projects ever approved in Illinois. Joliet is now a clear case study in how the Midwest’s data center market is evolving: massive land assemblies, utility-scale power requirements, front-loaded community concessions, increasingly organized local opposition, and regulators working to ensure that the costs of AI infrastructure are not shifted onto ratepayers. A Project Too Large to Call Routine The Joliet Technology Center is a campus-scale industrial platform built for the AI era. Plans call for 24 two-story buildings of roughly 144,500 square feet each, with total development estimated at approximately 6.9 million square feet and up to 1.8 GW of eventual capacity. That places the project firmly in the emerging “AI factory” category, e.g. far-removed from the incremental, metro-edge data center expansions that defined earlier growth cycles. The distinction is critical. AI-scale campuses operate on a different economic and technical model. Fiber access and metro proximity are no longer enough. These developments require large, contiguous power blocks, land to support phased substation and utility infrastructure, and a political framework capable of absorbing what is effectively heavy

AI is a Positive Catalyst for Grid Growth

Data centers, particularly those optimized for artificial intelligence workloads, are frequently characterized in public discourse as a disruptive threat to grid stability and ratepayer affordability. But behind-the-narrative as we are, the AI‑driven data center growth is simply illuminating pre‑existing systemic weaknesses in electric infrastructure that have accumulated over more than a decade of underinvestment in transmission, substations, and interconnection capacity. Over the same period, many utilities operated under planning assumptions shaped by slow demand growth and regulatory frameworks that incentivized incremental upgrades rather than large, anticipatory capital programs. As a result, the emergence of gigawatt‑scale computing campuses appears to be a sudden shock to a system that, in reality, was already misaligned with long‑term decarbonization, electrification, and digitalization objectives. Utilities have been asked to do more with aging grids, slow permitting, and chronically constrained capital, and now AI and cloud are finally putting real urgency — and real investment — behind modernizing that backbone. In that sense, large‑scale compute is not the problem; it is the catalyst that makes it impossible to ignore the problem any longer. We are at a moment when data centers, and especially AI data centers, are being blamed for exposing weaknesses that were already there, when in reality they are giving society a chance to fix a power system that has been underbuilt for more than a decade. Utilities have been asked to do more with aging grids, slow permitting, and limited investment, and now AI and cloud are finally putting real urgency — and real capital — behind modernizing that backbone. In that sense, data centers aren’t the problem; they are the catalyst that makes it impossible to ignore the problem any longer. AI Demand Provided a Long‑Overdue Stress Test The nature of AI workloads intensified this dynamic. High‑performance computing clusters concentrate substantial power

From Land Grab to Structured Scale: Kirkland & Ellis Explains How Capital, Power, and Deal Complexity Are Defining the AI Data Center Boom

The AI data center market is no longer defined by speed alone. For much of the past three years, capital moved aggressively into digital infrastructure, chasing land, power, and platform scale as generative AI workloads began to reshape demand curves. But as Melissa Kalka, M&A and private equity partner, and Kimberly McGrath, real estate partner at Kirkland & Ellis, explain on the latest episode of the Data Center Frontier Show, the industry is now entering a more complex and more consequential phase. The land grab is over. Execution has begun. Capital remains abundant, but it is no longer forgiving. From Capital Rush to Capital Discipline As noted by Kalka and McGrath, the period from roughly 2022 through 2025 marked a rapid acceleration in AI infrastructure investment. Take-private deals involving CyrusOne, QTS, and Switch signaled a structural shift, while hyperscale demand scaled from tens of megawatts to hundreds, and now toward gigawatt-class campuses. But the current phase is not defined by a pullback in capital. Instead, it reflects an expansion of investment pathways and a corresponding increase in scrutiny. “There’s actually more deal flow now,” Kalka notes, pointing to the growing range of entry points across the capital stack, including development vehicles, yield-oriented structures, and private credit. With more capital chasing larger and more complex opportunities, investors are evaluating not just platforms, but the full lifecycle of assets from early-stage development through stabilization and long-term hold. That shift has pulled capital earlier into the process, where risk is higher and less defined. Power availability, permitting, and execution timelines are now central to underwriting decisions. What Defines a “Bankable” Platform In this environment, the definition of a bankable data center platform has tightened. Execution history remains foundational. Investors are looking for consistent delivery, operational reliability, and clean contractual performance. But those factors alone

CoreWeave and Bell Canada Reset AI Data Center Scale

From GPU Cloud to AI Factory Operator In sum, CoreWeave is moving beyond its origins as a fast-scaling GPU cloud built on scarcity. The company is increasingly positioning itself as an AI infrastructure operator, where competitive advantage comes from integration across hardware, networking, cooling, platform software, workload orchestration, and early access to NVIDIA’s latest systems. That positioning has been reinforced by NVIDIA itself. In January, NVIDIA outlined a deeper alignment with CoreWeave focused on building AI factories, accelerating the procurement of land, power, and shell, and validating CoreWeave’s AI-native software and reference architecture. The partnership also includes deployment of multiple generations of NVIDIA infrastructure across CoreWeave’s platform, including Rubin systems, Vera CPUs, and BlueField data processing units, alongside a $2 billion equity investment. No simple vendor relationship, this is co-development around physical AI infrastructure. Bell Canada and the Rise of Sovereign AI Capacity Viewed through that lens, Bell Canada’s Saskatchewan announcement can be seen as part of the same structural shift. On March 16, Bell and the Government of Saskatchewan unveiled plans for a 300 MW AI Fabric data center in the Rural Municipality of Sherwood, outside Regina. CoreWeave is expected to anchor the site’s NVIDIA-based GPU infrastructure, extending its AI-native platform into a sovereign, hyperscale, power-dense environment. BCE described the project as its largest-ever investment in the province and said it is expected to become Canada’s largest purpose-built AI data center campus. Bell projects up to $12 billion (CDN) in long-term economic impact, along with at least 800 construction jobs and a minimum of 80 permanent roles once the site is operational. More importantly, Bell is explicitly framing the development as a foundation for domestic compute capacity, positioning AI infrastructure as a national asset tied to economic growth and technological sovereignty. That project extends Bell’s broader sovereign AI strategy.

From Reactor Designs to Real Projects: SMRs Enter the Execution Era as AI Power Demand Accelerates

The pattern emerging is clear. The SMR story is no longer about reactor design. Recent announcements are centered on permits, fuel, supply chains, financing, and customer traction, i.e. the factors that determine whether SMRs become a viable market or remain a technology narrative. The conversation has transitioned from technically compelling reactor concepts to the harder problem of industrial execution. Through the first quarter of 2026, and especially in March, vendors moved beyond partnership announcements to concrete progress in licensing, fuel access, supply-chain development, control systems, customer alignment, and capital formation. The distinction now is between companies building credible deployment pathways and those still positioned around long-dated opportunity. At a high level, these developments fall into three categories. First, regulatory progress: the most difficult and time-consuming milestone. Second, efforts to establish manufacturing and fuel ecosystems that can support repeatable, fleet-scale deployment. Third, a broad repositioning toward power-intensive industrial users, utilities, and increasingly data center–driven load growth. The result is an SMR market that looks less like a single competitive race and more like a set of parallel business models converging on the same objective: dispatchable, carbon-free power that can be financed and deployed with greater predictability than traditional gigawatt-scale nuclear. X-energy: Building a Commercial Path to Scale X-energy has emerged as one of the more credible commercialization stories in the SMR market, with recent moves spanning capital markets, customer development, and supply-chain expansion. Reuters reported on March 20 that the company has confidentially filed for an IPO, aiming to capitalize on renewed investor interest in nuclear and rising electricity demand tied to AI infrastructure. That filing followed closely on an agreement with Talen Energy to evaluate multiple four-unit Xe-100 deployments across U.S. power markets, as well as a MOU with Japan’s IHI to expand U.S.-Japan supply chain capabilities for the reactor.

DCF Poll: Data Centers and the Public Trust Gap

Matt Vincent is Editor in Chief of Data Center Frontier, where he leads editorial strategy and coverage focused on the infrastructure powering cloud computing, artificial intelligence, and the digital economy. A veteran B2B technology journalist with more than two decades of experience, Vincent specializes in the intersection of data centers, power, cooling, and emerging AI-era infrastructure. Since assuming the EIC role in 2023, he has helped guide Data Center Frontier’s coverage of the industry’s transition into the gigawatt-scale AI era, with a focus on hyperscale development, behind-the-meter power strategies, liquid cooling architectures, and the evolving energy demands of high-density compute, while working closely with the Digital Infrastructure Group at Endeavor Business Media to expand the brand’s analytical and multimedia footprint. Vincent also hosts The Data Center Frontier Show podcast, where he interviews industry leaders across hyperscale, colocation, utilities, and the data center supply chain to examine the technologies and business models reshaping digital infrastructure. Since its inception he serves as Head of Content for the Data Center Frontier Trends Summit. Before becoming Editor in Chief, he served in multiple senior editorial roles across Endeavor Business Media’s digital infrastructure portfolio, with coverage spanning data centers and hyperscale infrastructure, structured cabling and networking, telecom and datacom, IP physical security, and wireless and Pro AV markets. He began his career in 2005 within PennWell’s Advanced Technology Division and later held senior editorial positions supporting brands such as Cabling Installation & Maintenance, Lightwave Online, Broadband Technology Report, and Smart Buildings Technology. Vincent is a frequent moderator, interviewer, and keynote speaker at industry events including the HPC Forum, where he delivers forward-looking analysis on how AI and high-performance computing are reshaping digital infrastructure. He graduated with honors from Indiana University Bloomington with a B.A. in English Literature and Creative Writing and lives in southern New Hampshire with

Microsoft will invest $80B in AI data centers in fiscal 2025

And Microsoft isn’t the only one that is ramping up its investments into AI-enabled data centers. Rival cloud service providers are all investing in either upgrading or opening new data centers to capture a larger chunk of business from developers and users of large language models (LLMs). In a report published in October 2024, Bloomberg Intelligence estimated that demand for generative AI would push Microsoft, AWS, Google, Oracle, Meta, and Apple would between them devote $200 billion to capex in 2025, up from $110 billion in 2023. Microsoft is one of the biggest spenders, followed closely by Google and AWS, Bloomberg Intelligence said. Its estimate of Microsoft’s capital spending on AI, at $62.4 billion for calendar 2025, is lower than Smith’s claim that the company will invest $80 billion in the fiscal year to June 30, 2025. Both figures, though, are way higher than Microsoft’s 2020 capital expenditure of “just” $17.6 billion. The majority of the increased spending is tied to cloud services and the expansion of AI infrastructure needed to provide compute capacity for OpenAI workloads. Separately, last October Amazon CEO Andy Jassy said his company planned total capex spend of $75 billion in 2024 and even more in 2025, with much of it going to AWS, its cloud computing division.

John Deere unveils more autonomous farm machines to address skill labor shortage

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More Self-driving tractors might be the path to self-driving cars. John Deere has revealed a new line of autonomous machines and tech across agriculture, construction and commercial landscaping. The Moline, Illinois-based John Deere has been in business for 187 years, yet it’s been a regular as a non-tech company showing off technology at the big tech trade show in Las Vegas and is back at CES 2025 with more autonomous tractors and other vehicles. This is not something we usually cover, but John Deere has a lot of data that is interesting in the big picture of tech. The message from the company is that there aren’t enough skilled farm laborers to do the work that its customers need. It’s been a challenge for most of the last two decades, said Jahmy Hindman, CTO at John Deere, in a briefing. Much of the tech will come this fall and after that. He noted that the average farmer in the U.S. is over 58 and works 12 to 18 hours a day to grow food for us. And he said the American Farm Bureau Federation estimates there are roughly 2.4 million farm jobs that need to be filled annually; and the agricultural work force continues to shrink. (This is my hint to the anti-immigration crowd). John Deere’s autonomous 9RX Tractor. Farmers can oversee it using an app. While each of these industries experiences their own set of challenges, a commonality across all is skilled labor availability. In construction, about 80% percent of contractors struggle to find skilled labor. And in commercial landscaping, 86% of landscaping business owners can’t find labor to fill open positions, he said. “They have to figure out how to do

2025 playbook for enterprise AI success, from agents to evals

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More 2025 is poised to be a pivotal year for enterprise AI. The past year has seen rapid innovation, and this year will see the same. This has made it more critical than ever to revisit your AI strategy to stay competitive and create value for your customers. From scaling AI agents to optimizing costs, here are the five critical areas enterprises should prioritize for their AI strategy this year. 1. Agents: the next generation of automation AI agents are no longer theoretical. In 2025, they’re indispensable tools for enterprises looking to streamline operations and enhance customer interactions. Unlike traditional software, agents powered by large language models (LLMs) can make nuanced decisions, navigate complex multi-step tasks, and integrate seamlessly with tools and APIs. At the start of 2024, agents were not ready for prime time, making frustrating mistakes like hallucinating URLs. They started getting better as frontier large language models themselves improved. “Let me put it this way,” said Sam Witteveen, cofounder of Red Dragon, a company that develops agents for companies, and that recently reviewed the 48 agents it built last year. “Interestingly, the ones that we built at the start of the year, a lot of those worked way better at the end of the year just because the models got better.” Witteveen shared this in the video podcast we filmed to discuss these five big trends in detail. Models are getting better and hallucinating less, and they’re also being trained to do agentic tasks. Another feature that the model providers are researching is a way to use the LLM as a judge, and as models get cheaper (something we’ll cover below), companies can use three or more models to

OpenAI’s red teaming innovations define new essentials for security leaders in the AI era

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More OpenAI has taken a more aggressive approach to red teaming than its AI competitors, demonstrating its security teams’ advanced capabilities in two areas: multi-step reinforcement and external red teaming. OpenAI recently released two papers that set a new competitive standard for improving the quality, reliability and safety of AI models in these two techniques and more. The first paper, “OpenAI’s Approach to External Red Teaming for AI Models and Systems,” reports that specialized teams outside the company have proven effective in uncovering vulnerabilities that might otherwise have made it into a released model because in-house testing techniques may have missed them. In the second paper, “Diverse and Effective Red Teaming with Auto-Generated Rewards and Multi-Step Reinforcement Learning,” OpenAI introduces an automated framework that relies on iterative reinforcement learning to generate a broad spectrum of novel, wide-ranging attacks. Going all-in on red teaming pays practical, competitive dividends It’s encouraging to see competitive intensity in red teaming growing among AI companies. When Anthropic released its AI red team guidelines in June of last year, it joined AI providers including Google, Microsoft, Nvidia, OpenAI, and even the U.S.’s National Institute of Standards and Technology (NIST), which all had released red teaming frameworks. Investing heavily in red teaming yields tangible benefits for security leaders in any organization. OpenAI’s paper on external red teaming provides a detailed analysis of how the company strives to create specialized external teams that include cybersecurity and subject matter experts. The goal is to see if knowledgeable external teams can defeat models’ security perimeters and find gaps in their security, biases and controls that prompt-based testing couldn’t find. What makes OpenAI’s recent papers noteworthy is how well they define using human-in-the-middle

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