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Model Discovery of Partial Differential Equations

Abstract : Model discovery aims at autonomously discovering equations underlying a dataset. It is often approached as a sparse regression problem, selecting terms and constructing the unknown equation from a set of candidate feautures. In the case of partial differential equations, these features consist of higher-order derivatives, which are calculated using numerical differentiation. This makes it challenging to accurately calculate the candidate feature in experimental data, which is noisy and sparse. In this thesis we develop a neural network-based model discovery method, with a focus on discovering partial differential equations from noisy and sparse data. The foundation of our approach is a neural network which interpolates and denoises the data, and constrain the network to a given equation - a model known as physics informed neural networks. Simultaneously, we employ sparse regression to learn this constraining equation as the neural network is trained, yielding the underlying model. In the first part of this thesis we show that such an approach significantly improves the robustness of model discovery compared to unconstrained neural networks or other model discovery approaches. Improving on this, in the second part we present a modular framework, showing how these constrained networks can utilize any sparse regression algorithm. In the third part, we build upon the first two parts to achieve a fully differentiable model discovery algorithm by constraining a neural network with Sparse Bayesian Learning. Additionally, we introduce Conditional Normalizing Flows and show how they can be used to infer time-dependent probability distributions. Taken together, our work shows the importance of accurately modelling data for model discovery, and strongly establishes the argument for physics-constrained, neural network-based surrogates for model discovery of PDEs on experimental data.
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Submitted on : Thursday, March 10, 2022 - 2:38:08 PM
Last modification on : Tuesday, June 28, 2022 - 3:13:25 AM


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  • HAL Id : tel-03604486, version 1


Gert-Jan Both. Model Discovery of Partial Differential Equations. Machine Learning [stat.ML]. Université Paris sciences et lettres, 2021. English. ⟨NNT : 2021UPSLS088⟩. ⟨tel-03604486⟩



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