Generating synthetic data for neural operators (2401.02398v3)

Abstract: Recent advances in the literature show promising potential of deep learning methods, particularly neural operators, in obtaining numerical solutions to partial differential equations (PDEs) beyond the reach of current numerical solvers. However, existing data-driven approaches often rely on training data produced by numerical PDE solvers (e.g., finite difference or finite element methods). We introduce a "backward" data generation method that avoids solving the PDE numerically: by randomly sampling candidate solutions $u_j$ from the appropriate solution space (e.g., $H_0^1(\Omega)$), we compute the corresponding right-hand side $f_j$ directly from the equation by differentiation. This produces training pairs ${(f_j, u_j)}$ by computing derivatives rather than solving a PDE numerically for each data point, enabling fast, large-scale data generation consisting of exact solutions. Experiments indicate that models trained on this synthetic data generalize well when tested on data produced by standard solvers. While the idea is simple, we hope this method will expand the potential of neural PDE solvers that do not rely on classical numerical solvers to generate their data.

• The paper presents a synthetic data generation method that bypasses traditional numerical PDE solvers.
• It reverse engineers training data by inserting random functions into PDEs to compute efficient, diverse output samples.
• Empirical tests with the Fourier Neural Operator demonstrate accurate PDE solutions under various boundary conditions.

Introduction

Developments in deep learning have highlighted its potential to solve partial differential equations (PDEs), which are fundamental in modeling phenomena in science and technology. Neural operators are a category of deep learning models tailored to predict PDE solutions, and they can contend with complications such as high dimensionality that traditional numerical solvers struggle with. A critical bottleneck, however, lies in the need for extensive training data, typically generated by the very numerical solvers that neural operators aim to outperform. This paper proposes an inventive technique for synthetic training data production that eschews reliance on traditional numerical PDE solvers, potentially triggering a paradigm shift in the field of PDE-solving neural operators.

Data Generation Methodology

At the heart of the proposed methodology lies the reverse engineering of training data. Starting from random functions expected to be in the solution space of targeted PDEs (as per established theoretical mathematics), the authors conceptualize a synthetic dataset by inserting these functions into the PDEs and computing the corresponding right-hand side functions. To practitioners, this reversed scenario may seem like putting the cart before the horse, yet it brilliantly exploits the PDEs' properties to create a sheer volume of diverse training points. This strategy heavily relies on derivative computations, which are more computationally efficient compared to traditional PDE-solving techniques.

Empirical Findings and Applications

The paper validated the method by applying it to the Fourier Neural Operator (FNO), a leading-edge model for neural operator learning, with experiments targeting elliptic PDEs equipped with either Dirichlet or Neumann boundary conditions. The findings were encouraging: the synthetic data, tailored to the potential solution spaces, enabled FNO to predict PDE solutions with impressive accuracy. The researchers applied this approach to a spectrum of scenarios, from fixed matrices in the PDE equations to parametric matrices and tested the boundaries with semi-linear PDEs. In all cases, the method exhibited remarkable promise.

Conclusion and Prospects

Facilitating the training of neural PDE solvers without the need for classical numerical solvers is an exciting development, not least because it could address problems beyond the reach of current numerical methods. This data generation approach could reshape how the field views domain-specific data requirements, turning the focus toward a deeper theoretical understanding of the problems at hand. Although the paper concentrates on elliptic PDEs, the authors express confidence that the foundational ideation could extend to PDEs of different kinds. Looking to the future, the ability to harness synthetic, solver-independent data might just be the key to new heights in AI-driven computational science.