Fourier Neural Operators with Shock-Aware Loss for the Burgers Equation
DOI:
https://doi.org/10.29304/jqcsm.2026.18.22553Keywords:
Nonlinear PDEs, Operator learning, Gradient-based weighting, Scientific machine learning, Spatio-temporal modelingAbstract
The viscous Burgers equation is a benchmark for nonlinear conservation laws with shock formation, and poses a difficulty in obtaining accurate numerical solutions of non-smooth problems. We propose an operator-learning neural network to solve the one-dimensional viscous Burgers equation using Fourier transforms, which we term shock-aware Fourier neural operator (SA-FNO). The SA-FNO model employs a mixed loss function combining a mean squared error associated with the model output with a component that incorporates the effects of shocks via a shock-aware weighted loss function derived from the spatial gradient magnitude of the reference solution at each spatial location. The model learns an operator mapping from the initial conditions and time to the solution field. The operator allows for direct predictions across the entire spatio-temporal solution field. Our numerical results demonstrate that the SA-FNO model achieves a relative L2 error of 0.0248 for the entire time interval, an improvement of approximately 22% over a baseline FNO model trained with only shock-aware loss. We conclude that by training with a mixed shock-aware loss, the SA-FNO model can provide improved representation of non-smooth solutions, without modifying the underlying neural network architecture.
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