Generating synthetic data for neural operators
This work addresses a bottleneck for researchers in scientific machine learning by providing a faster data generation method, though it is incremental as it builds on existing neural operator frameworks.
The authors tackled the problem of generating training data for neural operators without relying on numerical PDE solvers by introducing a backward method that samples candidate solutions and computes corresponding right-hand sides via differentiation, producing exact solutions and enabling fast, large-scale data generation. Experiments showed that models trained on this synthetic data generalized well to data from standard solvers.
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., $H_0^1(Ω)$), 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.