Method of Manufactured Learning for Solver-free Training of Neural Operators

TL;DR

The paper introduces the Method of Manufactured Learning (MML), a solver-independent framework for training neural operators using analytically generated, physics-consistent datasets, enabling scalable and accurate learning without expensive data generation.

Contribution

MML replaces solver-based data with analytical synthesis for training neural operators, making the process solver-agnostic and scalable across physical systems.

Findings

Achieves high spectral accuracy on benchmark PDEs

Demonstrates strong generalization to unseen conditions

Reduces reliance on computationally expensive data generation

Abstract

Training neural operators to approximate mappings between infinite-dimensional function spaces often requires extensive datasets generated by either demanding experimental setups or computationally expensive numerical solvers. This dependence on solver-based data limits scalability and constrains exploration across physical systems. Here we introduce the Method of Manufactured Learning (MML), a solver-independent framework for training neural operators using analytically constructed, physics-consistent datasets. Inspired by the classical method of manufactured solutions, MML replaces numerical data generation with functional synthesis, i.e., smooth candidate solutions are sampled from controlled analytical spaces, and the corresponding forcing fields are derived by direct application of the governing differential operators. During inference, setting these forcing terms to zero restores the original governing equations, allowing the trained neural operator to emulate the true solution operator of the system. The framework is agnostic to network architecture and can be integrated with any operator learning paradigm. In this paper, we employ Fourier neural operator as a representative example. Across canonical benchmarks including heat, advection, Burgers, and diffusion-reaction equations. MML achieves high spectral accuracy, low residual errors, and strong generalization to unseen conditions. By reframing data generation as a process of analytical synthesis, MML offers a scalable, solver-agnostic pathway toward constructing physically grounded neural operators that retain fidelity to governing laws without reliance on expensive numerical simulations or costly experimental data for training.