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Solution of Advection Equation with Discontinuous Initial and Boundary Conditions via Physics-Informed Neural Networks

Omid Khosravi, Mehdi Tatari

TL;DR

The paper tackles solving the one-dimensional advection equation with discontinuous initial and boundary data using physics-informed neural networks (PINNs), formalized as $u_t + a(x,t,u)\,u_x = f(x,t,u)$. It introduces a random Fourier feature mapping to alleviate spectral bias and a two-stage training scheme that first learns the feature transform and then optimizes the network weights, complemented by adaptive loss weighting. Additional contributions include a median-filtering approach and a hard-output bound to stabilize the solution, along with an upwind-inspired loss for nonlinear advection to prevent excessive smoothing. The results demonstrate faster convergence and improved handling of discontinuities, with caveats such as potential overfitting and Gibbs-like artifacts, extending PINNs to hyperbolic problems with sharp features.

Abstract

In this paper, we investigate several techniques for modeling the one-dimensional advection equation for a specific class of problems with discontinuous initial and boundary conditions using physics-informed neural networks (PINNs). To mitigate the spectral bias phenomenon, we employ a Fourier feature mapping layer as the input representation, adopt a two-stage training strategy in which the Fourier feature parameters and the neural network weights are optimized sequentially, and incorporate adaptive loss weighting. To further enhance the approximation accuracy, a median filter is applied to the spatial data, and the predicted solution is constrained through a bounded linear mapping. Moreover, for certain nonlinear problems, we introduce a modified loss function inspired by the upwind numerical scheme to alleviate the excessive smoothing of discontinuous solutions typically observed in neural network approximations.

Solution of Advection Equation with Discontinuous Initial and Boundary Conditions via Physics-Informed Neural Networks

TL;DR

The paper tackles solving the one-dimensional advection equation with discontinuous initial and boundary data using physics-informed neural networks (PINNs), formalized as . It introduces a random Fourier feature mapping to alleviate spectral bias and a two-stage training scheme that first learns the feature transform and then optimizes the network weights, complemented by adaptive loss weighting. Additional contributions include a median-filtering approach and a hard-output bound to stabilize the solution, along with an upwind-inspired loss for nonlinear advection to prevent excessive smoothing. The results demonstrate faster convergence and improved handling of discontinuities, with caveats such as potential overfitting and Gibbs-like artifacts, extending PINNs to hyperbolic problems with sharp features.

Abstract

In this paper, we investigate several techniques for modeling the one-dimensional advection equation for a specific class of problems with discontinuous initial and boundary conditions using physics-informed neural networks (PINNs). To mitigate the spectral bias phenomenon, we employ a Fourier feature mapping layer as the input representation, adopt a two-stage training strategy in which the Fourier feature parameters and the neural network weights are optimized sequentially, and incorporate adaptive loss weighting. To further enhance the approximation accuracy, a median filter is applied to the spatial data, and the predicted solution is constrained through a bounded linear mapping. Moreover, for certain nonlinear problems, we introduce a modified loss function inspired by the upwind numerical scheme to alleviate the excessive smoothing of discontinuous solutions typically observed in neural network approximations.
Paper Structure (9 sections, 44 equations, 10 figures)

This paper contains 9 sections, 44 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Fourier Feature Mapping and Two-Stage Training
  3. Trainable Fourier Feature Mapping
  4. Adaptive Fourier Weighting
  5. Median Filtering and Bounded Approximation
  6. Median Filtering
  7. Bounding the Approximate Solution
  8. Upwind Loss Function
  9. Conclusion

Figures (10)

  • Figure 1: Evolution of the individual loss components when using random Fourier features and the two-stage training strategy.
  • Figure 2: Comparison of the loss components for two-stage training with and without adaptive Fourier weighting applied during the optimization of the Fourier-mapping parameters.
  • Figure 3: Filtered and unfiltered solutions at times $t=0$, $0.1$, and $0.15$.
  • Figure 4: Filtered and unfiltered solutions at times $t=0.6$, $0.7$, and $1$.
  • Figure 5: Schematic diagram of the proposed method
  • ...and 5 more figures