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The main obstacle is turbulence: chaotic swirling motion that occurs when fluid flows become sufficiently fast or complex. Instead of moving smoothly, the flow develops structures and fluctuations across many different scales, from large visible vortices down to tiny rapidly changing motions. Capturing all these scales at the same time requires enormous computational effort, making fully detailed simulations impractical for many real-world applications.

To address this challenge, PhD researcher developed new data-driven methods that improve both the efficiency and reliability of fluid simulations. He defended his PhD thesis at the Department of Mathematics and Computer Science on Tuesday, May 19.

Capturing the big picture

To make simulations more practical, scientists often use a technique called large eddy simulation, or LES. Rather than calculating every tiny swirl and fluctuation in a turbulent flow, LES concentrates on the larger motions that have the biggest impact on the overall behaviour of the fluid. The smaller motions are not simulated in full detail, but their effects are estimated using mathematical models.

In recent years, machine learning has emerged as a promising way to account for these missing effects directly from high quality data. However, while these models can significantly accelerate simulations, they often suffer from a key limitation: instability. Given that standard machine learning methods do not automatically follow the physics governing the behaviour of fluid flow, small errors can accumulate over time and eventually cause simulations to fail.

The role of physics in machine learning

In his PhD research, Van Gastelen addressed this problem by building physical principles directly into machine learning models.

Instead of relying only on data, the models are designed so that they naturally obey important physical properties of fluid flow, such as energy conservation and controlled energy dissipation. In other words, the physical rules are built directly into the structure of the model itself, rather than being corrected afterward.

As a result, the simulations remain stable by design. This prevents small errors from growing out of control during long simulations and leads to more reliable predictions over time. Across a wide range of test cases, these structure-preserving models consistently outperformed conventional approaches in both accuracy and robustness.

Faster alternatives with less data

In addition to neural network-based models models, Van Gastelen also explored simpler and more computationally efficient alternatives.

One approach improves simulations using a method known as the evolve-filter-relax framework, which can be added to existing simulation software without major modifications. In this approach, the simulation repeatedly calculates how the fluid moves, removes unstable small-scale disturbances, and then slightly adjusts the result before continuing to the next step.

Rather than training a large and computationally expensive machine learning model, this method learns how to improve these filtering and correction steps directly during the simulation. As a result, it avoids expensive training procedures and still performs well even when only limited data is available.

Reducing complexity without losing accuracy

Another part of the research focused on reduced-order modeling, a technique that simplifies complex fluid flows so they can be simulated much more quickly. Instead of calculating every detail of the flow, the method tries to capture only the most important motions that determine the overall behaviour of the fluid.

Traditional approaches do this by creating one simplified mathematical description for the entire flow. This works reasonably well for simple flows, but turbulent flows are far more difficult to describe in this way. Turbulence can vary strongly from one location to another and can change rapidly over time, meaning that one single overall description often cannot capture all important behaviour accurately.

To address this, Van Gastelen developed a new method that divides the flow into smaller regions and learns simplified descriptions for each region separately. By focusing on local behaviour instead of treating the entire flow as one system, the method can represent turbulent motion more accurately while still keeping the simulations computationally efficient.