Learning-based solutions to nonlinear hyperbolic PDEs: Empirical insights on generalization errors
This work addresses the problem of solving difficult PDEs for researchers in computational physics and machine learning, but it is incremental as it builds on existing methods with empirical insights.
The authors tackled learning weak solutions to nonlinear hyperbolic PDEs, which are challenging due to discontinuities, using a physics-informed Fourier Neural Operator (π-FNO) and found that generalization error grows linearly with input complexity, with improvements from a physics-informed regularizer for discontinuity prediction.
We study learning weak solutions to nonlinear hyperbolic partial differential equations (H-PDE), which have been difficult to learn due to discontinuities in their solutions. We use a physics-informed variant of the Fourier Neural Operator ($π$-FNO) to learn the weak solutions. We empirically quantify the generalization/out-of-sample error of the $π$-FNO solver as a function of input complexity, i.e., the distributions of initial and boundary conditions. Our testing results show that $π$-FNO generalizes well to unseen initial and boundary conditions. We find that the generalization error grows linearly with input complexity. Further, adding a physics-informed regularizer improved the prediction of discontinuities in the solution. We use the Lighthill-Witham-Richards (LWR) traffic flow model as a guiding example to illustrate the results.