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

TL;DR

The paper tackles the dependency of neural-operator training on solver-generated data by proposing the Method of Manufactured Learning (MML), a solver-free framework that constructs physics-consistent training data from analytical solution families. By sampling smooth candidate solutions and deriving exact forcing terms via the governing operator, MML embeds the PDE structure into training and enables zero-forcing inference to recover the true solution operator. The approach is architecture-agnostic and demonstrated with Fourier Neural Operators across four canonical time-dependent PDEs (heat, advection, Burgers, diffusion–reaction), achieving high spectral accuracy and robust generalization to unseen initial spectra. This solver-free data synthesis promises scalable pretraining, reduces discretization biases, and provides a principled pathway for exploring multi-physics and data-scarce regimes with physically grounded neural operators. The authors also release code and models publicly, highlighting practical impact for rapid prototyping and benchmarking in scientific ML.

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.

Figures (10)

• Figure 1: Method of Manufactured Learning (MML) workflow. MML constructs a self-supervised training framework by synthesizing solutions and residuals directly from the governing equations. A set of candidate functions is first introduced to automatically satisfy the prescribed initial and boundary conditions. These functions are then substituted into the governing equations, producing an analytically generated manufactured forcing term that defines the residual. This yields paired data without requiring simulations or experiments. A neural network is subsequently trained to learn the operator that maps residuals to solutions. Once trained, the manufactured forcing is set to zero, allowing the network to recover the desired physical solution of the original problem. In this way, MML removes the need for expensive pre-generated simulation or experimental data and provides a general pathway for training neural operators directly from physics.
• Figure 2: Time-slice comparisons for the one-dimensional heat equation under three unseen initial conditions. Columns correspond to the single-mode, two-mode, and three-mode initial configurations, while rows show the solution profiles at $t = 0$, $t = 0.5$, and $t = 1.0$. Higher-frequency modes are smoothed at the correct rate, and the predicted trajectories maintain spatial periodicity and temporal coherence throughout the full evolution.
• Figure 3: Spatio–temporal evolution of the one-dimensional heat equation under three unseen initial conditions. The corresponding relative $L^2$ errors for the predicted solution fields are $0.7158\%$ (single-mode), $2.361\%$ (two-mode), and $3.886\%$ (three-mode), demonstrating high quantitative accuracy and robust generalization to broadband initial conditions.
• Figure 4: Time-slice comparisons for the linear advection equation under three unseen initial conditions. Each column corresponds to a different initial condition, and rows show the solution at $t = 0$, $t = 0.5$, and $t = 1$. The MML-trained operator accurately captures the advective transport, preserving wave shapes and phase information across the entire time horizon.
• Figure 5: Spatio-temporal evolution of the one-dimensional advection equation predicted by the MML-trained operator for three unseen initial conditions. The operator faithfully reconstructs the downstream transport without artificial diffusion or distortion. The corresponding relative $L^2$ errors are $2.14\%$ (single-mode), $7.994\%$ (two-mode), and $12.37\%$ (three-mode), demonstrating accurate yet increasingly challenging reconstruction as spectral complexity increases.