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A Multi-Level Deep Framework for Deep Solvers of Partial Differential Equations

Yu Yang, Qiaolin He

TL;DR

The paper tackles the challenge of solving PDEs with neural solvers by addressing the slow learning of high-frequency components. It introduces a multi-level PINN framework comprising residual-driven Multi-Level Sampling and iterative Multi-Level Training, guided by a monitor function to concentrate samples in high-frequency regions, and leverages advanced optimizers (SOAP, SSBroyden) to accelerate convergence. A rigorous error-estimation analysis is provided, along with extensive numerical experiments on Poisson, Helmholtz, sharp-interface Poisson, 3D interfaces, and Prandtl equations, demonstrating superior accuracy and robustness over traditional PINN strategies. The approach holds promise for efficient, high-fidelity neural PDE solvers in problems with strong frequency content or low regularity.

Abstract

In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling method proposed in this paper is first employed to obtain new training points, so that these points become increasingly concentrated in computational regions corresponding to high-frequency components. Then, the generalization ability of deep neural networks are utilized to update the PDEs for the next level of training based on the results from all previous levels. Rigorous mathematical proofs and detailed numerical experiments are employed to demonstrate the effectiveness of the proposed method.

A Multi-Level Deep Framework for Deep Solvers of Partial Differential Equations

TL;DR

The paper tackles the challenge of solving PDEs with neural solvers by addressing the slow learning of high-frequency components. It introduces a multi-level PINN framework comprising residual-driven Multi-Level Sampling and iterative Multi-Level Training, guided by a monitor function to concentrate samples in high-frequency regions, and leverages advanced optimizers (SOAP, SSBroyden) to accelerate convergence. A rigorous error-estimation analysis is provided, along with extensive numerical experiments on Poisson, Helmholtz, sharp-interface Poisson, 3D interfaces, and Prandtl equations, demonstrating superior accuracy and robustness over traditional PINN strategies. The approach holds promise for efficient, high-fidelity neural PDE solvers in problems with strong frequency content or low regularity.

Abstract

In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling method proposed in this paper is first employed to obtain new training points, so that these points become increasingly concentrated in computational regions corresponding to high-frequency components. Then, the generalization ability of deep neural networks are utilized to update the PDEs for the next level of training based on the results from all previous levels. Rigorous mathematical proofs and detailed numerical experiments are employed to demonstrate the effectiveness of the proposed method.
Paper Structure (19 sections, 3 theorems, 50 equations, 16 figures, 3 tables, 2 algorithms)

This paper contains 19 sections, 3 theorems, 50 equations, 16 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary work
  3. Problem formulation
  4. Introduction to PINNs
  5. Main results
  6. Methods
  7. Multi-level Sampling
  8. Multi-level Training
  9. Optimization methods
  10. Error Estimation
  11. Numerical Experiments
  12. Symbols and parameter settings
  13. Two-dimensional Poisson equation
  14. Two-dimensional Helmholtz equation
  15. Poisson Equation with Sharp Solutions
  16. ...and 4 more sections

Key Result

Theorem 1

For any $\boldsymbol{u}_1 \in \mathbb{H}$ and $\forall k \geq 2$, there exists a sequence of functions $\left\{\boldsymbol{u}_i \right\}_{i=1}^k \subset \mathbb{H}$ such that: (i) $\sum_{i=1}^k \boldsymbol{u}_i = \boldsymbol{u}^*$; (ii) $\left \{\Vert\sum_{i=1}^j \boldsymbol{u}_i -\boldsymbol{u}^* \

Figures (16)

  • Figure 1: This figure illustrates a comparison between a two-level multigrid method and a multi-level framework. In the multigrid method, computation is first performed on the fine grid, after which the residual is restricted to the coarse grid. An error equation is then solved on the coarse grid to approximate the low-frequency error components that are difficult to eliminate on the fine grid. Finally, the correction obtained on the coarse grid is interpolated back to the fine grid to update the solution there. In the multi-level framework, training is first conducted on uniformly distributed sample points. The generalization capability of the trained neural network and a multi-level sampling strategy are then leveraged to update the set of sample points. A new equation is subsequently solved on these new sample points. Ultimately, the predicted solutions from the different neural networks are summed to yield the final predicted solution for the original equation (for a detailed introduction of our method, please refer to Section \ref{['sec:Methods']}).
  • Figure 2: Flow chart of our framework.
  • Figure 3: Here are the heatmaps of the true solutions Eq \ref{['eq:2d_Peaksolution']} and Eq \ref{['eq:Helmholtz_solution']}.
  • Figure 4: The sampling points at different levels for the two-dimensional Poisson equation with the solution Eq \ref{['eq:2d_Peaksolution']}. This figure shows the behavior of $40 \times 40$ points in $[-0.5,0.5]^2$.
  • Figure 5: The numerical result of different levels for the two-dimensional Poisson equation with the solution Eq \ref{['eq:2d_Peaksolution']}.
  • ...and 11 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Definition 1
  • Definition 2
  • Theorem 2
  • proof