Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks
This addresses reliability issues in PINNs for researchers and practitioners in scientific machine learning, offering an incremental improvement over existing methods.
The paper tackles the problem of training failures in Physics Informed Neural Networks (PINNs) due to multi-scale dynamics and objective conflicts, proposing an inverse-Dirichlet weighting strategy that achieves orders of magnitude improvement in accuracy and convergence, and prevents catastrophic forgetting in inverse modeling.
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 $\boldsymbolε$-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.