HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions

TL;DR

HyPINO tackles zero-shot generalization for multi-physics PDE solving by learning a hypernetwork that generates PINN parameters conditioned on PDE specifications and training on MMS-supervised and physics-informed data. It introduces an iterative residual refinement to form ensembles, improving accuracy without retraining; it achieves strong zero-shot performance across seven benchmarks and improves fine-tuning efficiency when used as initialization. The approach demonstrates a scalable pathway toward general-purpose, data-efficient neural operators for nonlinear, high-dimensional PDEs, with code publicly available.

Abstract

We present HyPINO, a multi-physics neural operator designed for zero-shot generalization across a broad class of PDEs without requiring task-specific fine-tuning. Our approach combines a Swin Transformer-based hypernetwork with mixed supervision: (i) labeled data from analytical solutions generated via the Method of Manufactured Solutions (MMS), and (ii) unlabeled samples optimized using physics-informed objectives. The model maps PDE parameterizations to target Physics-Informed Neural Networks (PINNs) and can handle linear elliptic, hyperbolic, and parabolic equations in two dimensions with varying source terms, geometries, and mixed Dirichlet/Neumann boundary conditions, including interior boundaries. HyPINO achieves strong zero-shot accuracy on seven benchmark problems from PINN literature, outperforming U-Nets, Poseidon, and Physics-Informed Neural Operators (PINO). Further, we introduce an iterative refinement procedure that treats the residual of the generated PINN as "delta PDE" and performs another forward pass to generate a corrective PINN. Summing their contributions and repeating this process forms an ensemble whose combined solution progressively reduces the error on six benchmarks and achieves a >100x lower $L_2$ loss in the best case, while retaining forward-only inference. Additionally, we evaluate the fine-tuning behavior of PINNs initialized by HyPINO and show that they converge faster and to lower final error than both randomly initialized and Reptile-meta-learned PINNs on five benchmarks, performing on par on the remaining two. Our results highlight the potential of this scalable approach as a foundation for extending neural operators toward solving increasingly complex, nonlinear, and high-dimensional PDE problems. The code and model weights are publicly available at https://github.com/rbischof/hypino.

Figures (14)

• Figure 1: Overview of the HyPINO pipeline. (a) Training data includes supervised samples from MMS and unsupervised physics-informed samples without ground truth. (b) PDEs are encoded as multi-channel and vector-based inputs and processed by HyPINO to produce task-specific PINN weights. (c) The predicted PINN maps spatial coordinates to the solution field. (d) Training combines physics-informed residual losses as well as supervised losses for MMS data. (e) At inference, HyPINO enables zero-shot prediction for unseen PDEs. (f) Downstream adaptation includes iterative refinement using residual corrections or optional, task-specific fine-tuning.
• Figure 2: Sample generated via MMS with sampled operator $\mathcal{L}[u] = -0.31 u_{xx} - 0.15 u_y$ and sampled boundaries $\partial \Omega$: (a) Dirichlet boundary, (b) Dirichlet condition, (c) Neumann boundary, (d) Neumann condition, (e) source term, and (f) analytical solution.
• Figure 3: Effect of iterative refinement on HyPINO predictions across benchmarks. MSE (left) and relative error (right) as functions of refinement iterations. Relative error at iteration $i$ is the ratio of MSE at iteration $i$ to that at iteration 0.
• Figure 4: Convergence on the 1D Heat Equation (HT) for randomly initialized PINNs (blue), Reptile-initialized PINNs (orange), and HyPINO-initialized PINNs.
• Figure 5: Parameterization of the 1D Heat PDE.