Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains

TL;DR

The paper tackles PDEs with highly oscillatory solutions on complex domains, where traditional discretization and black-box neural methods struggle. It introduces a multi-scale Finite Expression Method (FEX) with three innovations: symbolic spectral composition to learn multiscale frequencies, a redesigned linear input layer to boost expressivity, and an eigenvalue formulation to handle eigenproblems. The method yields explicit, interpretable closed-form expressions and demonstrates superior accuracy on Poisson, Poisson–Boltzmann, Helmholtz-type problems, and Laplace eigenvalue problems, particularly on domains with holes, outperforming existing neural solvers. While effective, the approach incurs computational cost in the operator-search loop and faces challenges in scaling to larger PDE systems and handling widely separated spectral components.

Abstract

Solving partial differential equations (PDEs) with highly oscillatory solutions on complex domains remains a challenging and important problem. High-frequency oscillations and intricate geometries often result in prohibitively expensive representations for traditional numerical methods and lead to difficult optimization landscapes for machine learning-based approaches. In this work, we introduce an enhanced Finite Expression Method (FEX) designed to address these challenges with improved accuracy, interpretability, and computational efficiency. The proposed framework incorporates three key innovations: a symbolic spectral composition module that enables FEX to learn and represent multiscale oscillatory behavior; a redesigned linear input layer that significantly expands the expressivity of the model; and an eigenvalue formulation that extends FEX to a new class of problems involving eigenvalue PDEs. Through extensive numerical experiments, we demonstrate that FEX accurately resolves oscillatory PDEs on domains containing multiple holes of varying shapes and sizes. Compared with existing neural network-based solvers, FEX achieves substantially higher accuracy while yielding interpretable, closed-form solutions that expose the underlying structure of the problem. These advantages, often absent in conventional finite element, finite difference, and black-box neural approaches, highlight FEX as a powerful and transparent framework for solving complex PDEs.

Figures (16)

• Figure 1: Construction of expressions using binary trees. Each node is either a binary or unary operator. Beginning with the basic unary and binary trees, mathematical expressions can be built by performing computation recursively. Each tree node is either a binary operator or a unary operator that takes value from the corresponding binary or unary set. The binary set can be $\mathbb{B}:=\{+,-,\times,\cdots\}$. The unary set can be $\mathbb{U}:=\{\sin,\exp, \log, \text{Id}, (\cdot)^2, \int\cdot\text{d} x_i, \frac{\partial\cdot}{\partial x_i}, \cdots\}$, which contains elementary functions (e.g., polynomial and trigonometric function), and even integration or differentiation operators. Here "Id" denotes the identity map. Note that if an integration or a derivative is used in the expression, the operator can be applied using a numerical method.
• Figure 2: A flowchart outlining the FEX-PG algorithm: The search loop consists of four key components: score computation, operator sequence generation, controller updates, and candidate optimization.
• Figure 3: A detailed illustration of the expression generation block from the algorithm flowchart in Figure \ref{['fig:FEXLoop']}.
• Figure 4: A comparison of the old and new input layer. Here $x\in \mathbb{R}^d$ is the input. Each component of $\textbf{x}$ is multiplied by weight $\alpha_i$. A unary operator $u_m$ is sampled from the set of unary operators in both cases, but due to the changes in the input layer structure, the resulting expressions are quite different. The old input layer combines the terms $u_m(x_i)$ with addition always, resulting in a sum. The new layer uses a set of binary operators (in practice, addition and multiplication), resulting in either a sum or product, depending on which operator is sampled. This is combined with the new weights, allowing for far greater expressivity. Both outcomes are shown to emphasize the expanded possibilities given by this reformulation.
• Figure 5: An example of the use of the multi-scaled periodic functions, where multiple scales of sines are composed into a candidate solution $\tilde{u}(\textbf{x})$.