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

NeurIPS 2025 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Physics-informed neural network training · 0 citations (OpenAlex)

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Proposes HyPINO, a Swin-Transformer hypernetwork that maps 2D linear PDE specifications to PINN weights, trained on mixed MMS-supervised and physics-only samples, with residual-driven iterative refinement for zero-shot solving.

Problem

Zero-shot generalization of neural operators across diverse parametric linear 2D PDEs (varying operators, source terms, geometries, mixed Dirichlet/Neumann boundaries) without task-specific fine-tuning, while avoiding dependence on large labeled simulation datasets.

Approach

A hypernetwork (Swin Transformer encoder with FiLM operator conditioning and attention pooling) generates the full weights of a target PINN from a grid-based PDE parameterization, trained with a hybrid of MMS-supervised samples and physics-informed unlabeled samples; residuals of the generated PINN are fed back as a "delta PDE" to produce corrective PINNs that are summed into an ensemble.

Key results

• HyPINO^10 achieves the best MSE on most benchmarks (e.g., HT 8.0e-5 vs PINO 1.4e-2 zero-shot baseline), with state-of-the-art on all but two benchmarks after 10 refinement rounds.
• Zero-shot HyPINO average rank 2.00 versus 3.00 for U-Net, 2.86 for Poseidon, and 2.14 for PINO.
• HyPINO-initialized PINNs converge faster and to lower final error than random and Reptile initializations on 4 of 7 benchmarks, matched on 2, and underperformed on 1.

Measured effects reported in the paper

Surprising findings

• Iterative residual refinement also improves PINO (PINO^3, PINO^10), suggesting it is a generic test-time enhancement.
• Zero-shot HyPINO (single pass) is not uniformly better than baselines; it underperforms PINO on several benchmarks such as PS-C and WV before refinement.

Limitations

• Restricted to linear 2D PDEs with spatially uniform coefficients.
• Unsupervised physics-only samples may be ill-posed; well-posedness is only heuristically ensured.
• Output space is less structured than grid-based baselines, and zero-shot performance trails on some benchmarks (e.g., HZ-G, WV).

Experimental setup

Datasets: synthetic MMS-generated PDEs synthetic unsupervised physics-informed PDEs PINN benchmark problems (DeepXDE, PINNacle)

Benchmarks: HT HZ HZ-G PS-C PS-L PS-G WV

Metrics: MSE SMAPE average rank relative error vs refinement iterations steps to reach target MSE in fine-tuning

Baselines: U-Net Poseidon-B PINO PINO^3 PINO^10 Reptile

Builds on: HyperPINN Poseidon PINO Reptile Swin Transformer

Closest papers by meaning

• RoPINN: Region Optimized Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.89)
• Neuro-Spectral Architectures for Causal Physics-Informed Networks (NeurIPS 2025 · similarity 0.88)
• RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains (NeurIPS 2025 · similarity 0.88)
• Per-Loss Adapters for Gradient Conflict in Physics-Informed Neural Networks (NeurIPS 2026 · similarity 0.88)
• Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.88)
• PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs (NeurIPS 2024 · similarity 0.88)