A Least-Squares-Based Neural Network (LS-Net) for Solving Linear Parametric PDEs

TL;DR

This work addresses efficient solution of linear parametric PDEs by introducing LS-Net, a least-squares neural solver that builds a parametric solution in a separated form $u^{p,\alpha}(x)=\mathbf{c}^{p,\alpha}\cdot\mathbf{u}^{\alpha}(x)$ where the NN outputs the basis $\mathbf{u}^{\alpha}$. A per-parameter LS problem determines coefficients $\mathbf{c}^{p,\alpha}$ for each $p$, while training optimizes the neural basis across the parameter space via the parametric loss $\mathcal{L}_{\mu}(\alpha)$; two residual-based losses, PINN (strong form) and Deep Fourier Residual (weak form), are explored. A universal-approximation–type theorem guarantees the existence of a sufficiently large network that achieves prescribed accuracy, and numerical experiments on a damped oscillator, a 1D Helmholtz problem with impedance BC, and a 2D transmission problem demonstrate accurate parametric solutions and discretization-invariance. Limitations include the LS cost per parameter, which is mitigated by a pretraining step that enables a parametric LS system after training; future work includes extending to nonlinear PDEs, optimizing the LS evaluation, and refining boundary- and interface-aware NN architectures.

Abstract

Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It utilizes a separated representation form for the parametric PDE solution via a deep neural network and a least-squares solver. In this approach, the output of the deep neural network consists of a vector-valued function, interpreted as basis functions for the parametric solution space, and the least-squares solver determines the optimal solution within the constructed solution space for each given parameter. The LS-Net method requires a quadratic loss function for the least-squares solver to find optimal solutions given the set of basis functions. In this study, we consider loss functions derived from the Deep Fourier Residual and Physics-Informed Neural Networks approaches. We also provide theoretical results similar to the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can theoretically approximate solutions of parametric PDEs with the desired accuracy. We illustrate the LS-net method by solving one- and two-dimensional problems. Numerical results clearly demonstrate the method's ability to approximate parametric solutions.

Figures (13)

• Figure 1: Schematic of the LS-Net architectures for solving parametric PDEs.
• Figure 2: NN architecture.
• Figure 3: Relative $L^2$-errors (in $\%$) of the LS-Net solutions and their first and second derivatives for the damped harmonic oscillator problem. We display the errors after solving the corresponding LS problem, (a) before and (b) after training the NN. Note the difference in the ranges of the color bars.
• Figure 4: Distribution of the relative $L^2$-errors (in $\%$) for the LS-Net solution and their first and second derivatives of the damped harmonic oscillator problem. Results include solving the LS problem before and after training the NN.
• Figure 5: The exact and the LS-Net solutions of the damped harmonic oscillator problem, illustrate the three damping behaviors: overdamped, critically damped, and underdamped.

Theorems & Definitions (13)

• Remark 2.1
• Theorem 2.2
• Remark 3.1
• Remark 3.2
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Theorem 4.3
• proof