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Parameterized Physics-informed Neural Networks for Parameterized PDEs
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Proposes parameterized PINNs (P²INNs) that encode PDE parameters with a separate encoder so one model solves a family of parameterized CDR and Helmholtz PDEs, with SVD-based fast fine-tuning.
Problem
PINNs must be retrained from scratch for each new parameter setting of parameterized PDEs and struggle on hard regimes (large convection/reaction coefficients, known failure modes).
Approach
A modular network with separate encoders for spatiotemporal coordinates and PDE parameters, concatenated into a manifold network that outputs the solution, trained jointly on mini-batches with multiple PDE parameter configurations; the decoder layers are SVD-decomposed so only diagonal singular-value scalings are fine-tuned.
Key results
- On reaction equations with coefficient range 1 to 20, P²INN reaches absolute error 0.0042 versus 0.4320 for the best PINN baseline (PINN-R), a roughly 100x reduction.
- Across CDR equation types and ranges, P²INN improves over the best baseline by about 33% to 99% in absolute error as reported in Table 2.
- On 2D Helmholtz, P²INN keeps absolute error around 0.023 to 0.032 across seen and unseen a values, while PINN and PINN-R errors reach 1.9 and 1.9 respectively.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use P$^2$INN instead of the strongest listed baseline | 1D CDR equations, coefficient range 1~20 · Reaction equations | Abs. err.: 0.4320 -> 0.0042 | Helped |
| use P$^2$INN instead of PINN-seq2seq | 1D CDR equations, coefficient range 1~20 · Diffusion equations | Abs. err.: 0.3346 -> 0.1916 | Helped |
| use P$^2$INN instead of PINN-seq2seq | 1D CDR equations, coefficient range 1~20 · Conv.-Diff. equations | Abs. err.: 0.0645 -> 0.0622 | Helped |
| use P$^2$INN instead of PINN-R | 2D Helmholtz · a=2.75 unseen | Abs. err.: 1.4000 -> 0.0263 | Helped |
| feed PDE parameters as coordinates without a separate parameter encoder (ablation) | P$^2$INN · Reaction equations, coefficient 1~20 | Abs. err.: 0.0042 -> 0.8975 | Hurt |
| feed PDE parameters as coordinates without a separate parameter encoder (ablation) | P$^2$INN · Reaction equations, coefficient 1~5 | Abs. err.: 0.0015 -> 0.0083 | Hurt |
| increase PDE-parameter encoder dimension from 80 to 320 (ablation) | P$^2$INN · Convection equations, coefficient 1~5 | Abs. err.: 0.0030 -> 0.0023 | Helped |
| increase PDE-parameter encoder dimension from 160 to 320 (ablation) | P$^2$INN · Reaction equations, coefficient 1~5 | Abs. err.: 0.0016 -> 0.0061 | Hurt |
| use SVD modulation rather than updating all parameters (ablation) | P$^2$INN fine-tuning · Convection equations | Abs. err.: 0.1959 -> 0.0138 | Helped |
| use SVD modulation rather than shift modulation (ablation) | P$^2$INN fine-tuning · Reaction equations | Abs. err.: 0.0095 -> 0.0089 | Helped |
| use SVD modulation rather than shift modulation (ablation) | P$^2$INN fine-tuning · Diffusion equations | Rel. err.: 0.1785 -> 0.1518 | Hurt |
| use P$^2$INN instead of PINN | PDE failure modes · Convection equation β=30 | Abs. err.: 0.6132 -> 0.0910 | Helped |
Surprising findings
- Shift modulation in PINNs did not give significant improvement over the pretrained model, motivating the SVD-based alternative.
- On the diffusion equations with coefficient range 1 to 5, PINN and P²INN perform similarly (0.97% improvement), and P²INN is worse than PINN on some initial-condition settings in the appendix.
Limitations
- The authors note that for a single trivial equation, existing PINN methods are more efficient.
- Evaluation is restricted to 1D CDR equations and 2D Helmholtz equations with analytic or numerical reference solutions.
- Some diffusion settings show P²INN with worse max error or explained variance than baselines.
Experimental setup
Datasets: 1D convection-diffusion-reaction equations (synthetic, generated with Strang splitting) 2D Helmholtz equations (analytic solutions)
Benchmarks: 1D convection-diffusion-reaction equations 2D Helmholtz equation
Metrics: L2 absolute error L2 relative error max error explained variance score
Baselines: PINN PINN-R PINN-seq2seq PINN-P MAML Reptile
Builds on: physics-informed neural networks SVD-PINNs MAML Reptile
Closest papers by meaning
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.90)
- Sub-Sequential Physics-Informed Learning with State Space Model (ICML 2025 · similarity 0.89)
- PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks (ICLR 2024 · similarity 0.89)
- Efficient Error Certification for Physics-Informed Neural Networks (ICML 2024 · similarity 0.88)
- Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.88)
- PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations (ICLR 2025 · similarity 0.88)
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