How ML Can Solve Partial Differential Equations

This piece, by Onno Berkan, was published on 11/05/24. The original text, by Brunton & Kutz, was published by Nature Computational Science on 06/28/2024.

This UW-Seattle study explores how machine learning can improve our understanding and solving of partial differential equations (PDEs), which describe various physical phenomena such as fluid flow and heat transfer. Traditionally, deriving PDEs relied heavily on established scientific principles. However, the growing availability of data has opened new avenues for discovering these equations from scratch with machine learning techniques. 

Three critical areas are highlighted: first, finding new governing equations and approximations tailored for complex systems; second, developing effective representations and models that simplify higher-dimensional systems, making them easier to analyze; and third, enhancing existing numerical algorithms for solving PDEs more efficiently.

One of the critical methods discussed is the PDE-FIND algorithm, which enables researchers to derive PDEs directly from data. It does this by examining how changes in a system's state over time can be represented as a combination of different effects. This method showcases how machine learning can play a role in scientific discovery by identifying new physical relationships.

The study also highlights advancements in converting complicated nonlinear systems into linear representations, making them more straightforward. Furthermore, the research discusses the importance of creating interpretable and generalizable models rather than merely providing results that work only within a limited context. In this way, machine learning isn’t just about more data but about understanding the underlying principles governing the phenomena.

Lastly, the study emphasizes that the integration of machine learning with traditional sciences is still in its early stages, particularly regarding complex biological and engineered systems. By combining these fields, researchers hope to discover new insights that were previously unattainable, paving the way for innovations across various scientific domains.

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