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Physics-Informed Machine Learning

Scalable & Provable Correct Manifold Learning of Latent Dynamics

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Sectorplan Bèta under the focus area Mathematics of Computational Science


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Description:

Many processes can be described by dynamical systems. To observe the current behavior and predict the future states of these processes, we need fast and provably accurate numerical solvers. When the dimensionality of a dynamical system is limited, there are many suitable options to use as solvers. However, the dimensionality grows drastically when the needed level of details increases. This limits the number of solvers available and reduces the efficiency of solving the dynamical systems.

In reduced order modelling, the goal is to accelerate finding the solution of the dynamical systems by reducing their dimensionality, yet without significantly compromising the accuracy. The key idea is to change the coordinates of the dynamical system and project down into a low-dimensional latent space. By smartly changing coordinates, this reduced space contains most if not all of the relevant information of the original system. In practice, this change in coordinates and projection are often carried out through the proper orthogonal decomposition of solution snapshots. Although this works for many problems, there are still plenty of cases for which it does not, especially in computational physics. In this project, we will augment the model reduction method by using auto-encoders to find a suitable latent space. By making use of novel algorithms and appropriate losses for training the auto-encoders, we aim to find smaller latent spaces that allow us to solve higher-dimensional dynamics efficiently, and simultaneously prove the solution accuracy.

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