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Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks

Maddu, S.; Sturm, D.; Müller, C. L.; Sbalzarini, I. F.

We characterize and remedy a failure mode that may arise from multi-scale dynamics with scale
imbalances during training of deep neural networks, such as Physics Informed Neural Networks
(PINNs). PINNs are popular machine-learning templates that allow for seamless integration of
physical equation models with data. Their training amounts to solving an optimization problem over
a weighted sum of data-fidelity and equation-fidelity objectives. Conflicts between objectives can
arise from scale imbalances, heteroscedasticity in the data, stiffness of the physical equation, or from
catastrophic interference during sequential training. We explain the training pathology arising from
this and propose a simple yet effective inverse-Dirichlet weighting strategy to alleviate the issue. We
compare with Sobolev training of neural networks, providing the baseline of analytically
$\epsilon$-optimal training. We demonstrate the effectiveness of inverse-Dirichlet weighting in various applications,
including a multi-scale model of active turbulence, where we show orders of magnitude improvement
in accuracy and convergence over conventional PINN training. For inverse modeling using sequential
training, we find that inverse-Dirichlet weighting protects a PINN against catastrophic forgetting.

Keywords: physics informed neural networks; sobolev training; computational physics; deep learning; data-driven modeling; multi-objective optimization; catastrophic forgetting; active turbulence

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Permalink: https://www.hzdr.de/publications/Publ-33335
Publ.-Id: 33335