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

• URL: http://arxiv.org/abs/2509.05117v1
• Date: Fri, 05 Sep 2025 13:59:25 GMT
• Title: HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions
• Authors: Rafael Bischof, Michal Piovarči, Michael A. Kraus, Siddhartha Mishra, Bernd Bickel,
• Abstract summary: We present HyPINO, a multi-physics neural operator designed for zeroshot generalization across a broad class of parametric PDEs.<n>The model maps PDE parametrizations to target Physics-Informed Neural Networks (PINNs) and can handle linear elliptic, hyperbolic, and parabolic equations in two dimensions.<n>HyPINO achieves strong zero-shot accuracy on seven benchmark problems from PINN, outperforming U-Nets, literature, and Physics-Informed Neural Operators (PINO)
• Score: 16.904297509040777
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present HyPINO, a multi-physics neural operator designed for zero-shot generalization across a broad class of parametric 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 parametrizations 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 compares the physics of the generated PINN to the requested PDE and uses the discrepancy to generate a "delta" PINN. Summing their contributions and repeating this process forms an ensemble whose combined solution progressively reduces the error on six benchmarks and achieves over 100x gain in average $L_2$ 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 with significantly improved accuracy and reduced computational cost.

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