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Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers
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Proposes Φ-ROM, a reduced-order model that trains a latent dynamics network directly against time derivatives from differentiable PDE solvers (via hyper-reduced least squares), yielding better generalization and extrapolation than DINo and other physics-informed ROMs.
Problem
Data-driven nonlinear reduced-order models of time-dependent parameterized PDEs drift away from the governing physics, generalizing poorly to unseen parameters and beyond the training horizon, because numerical solvers are excluded from training.
Approach
Embed a differentiable numerical solver in ROM training: project the solver's time derivatives into latent space via a hyper-reduced least-squares problem (through the INR decoder Jacobian) and train the latent dynamics network on these physics-based targets, with gradients flowing back into the decoder and latent codes.
Key results
- On Diffusion extrapolation, Φ-ROM achieves RNMSE 0.034 vs DINo 0.051 and PINN-ROM 0.042.
- On N-S test-IC extrapolation, Φ-ROM is more than four times better than DINo (0.373 vs 1.543 RNMSE).
- On LBM with unseen Reynolds numbers, Φ-ROM keeps extrapolation error ~0.18 while DINo reaches 0.566; with 2% sparse training DINo degrades to 0.412 on interpolation parameters versus Φ-ROM 0.188.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train latent dynamics using solver-derived physics instead of data-driven latent integration | Φ-ROM vs. DINo · N-S, test ICs, temporal extrapolation | RNMSE: 1.543 -> 0.373 | Helped |
| Train latent dynamics using solver-derived physics instead of data-driven latent integration | Φ-ROM vs. DINo · KdV, test ICs, temporal extrapolation | RNMSE: 0.728 -> 0.486 | Helped |
| Train latent dynamics using solver-derived physics instead of data-driven latent integration | Φ-ROM vs. DINo · LBM, test parameters, temporal extrapolation | RNMSE: 0.566 -> 0.180 | Helped |
| Train latent dynamics using solver-derived physics instead of data-driven latent integration | Φ-ROM vs. DINo · Burgers', [T_tr, T_te] | RNMSE: 0.060 -> 0.028 | Helped |
| Use solver-informed dynamics training rather than DINo training | Φ-ROM vs. DINo · N-S, test ICs, interpolation | RNMSE: 0.580 -> 0.170 | Helped |
| Use solver-informed dynamics training rather than DINo training | Φ-ROM vs. DINo · LBM, test parameters, interpolation | RNMSE: 0.457 -> 0.115 | Helped |
| Use solver-informed dynamics training rather than DINo training | Φ-ROM vs. DINo · Diffusion, [T_tr, T_te] | RNMSE: 0.051 -> 0.034 | Helped |
| Use solver-informed dynamics training rather than PINN-ROM | Φ-ROM vs. PINN-ROM · Burgers', [T_tr, T_te] | RNMSE: 0.348 -> 0.028 | Helped |
| Use solver-informed dynamics training rather than FD-CROM | Φ-ROM vs. FD-CROM · Burgers', [T_tr, T_te] | RNMSE: 0.044 -> 0.028 | Helped |
| Use solver-informed dynamics training rather than FD-CROM | Φ-ROM vs. FD-CROM · Burgers', [0, T_tr] | RNMSE: 0.001 -> 0.021 | Hurt |
| Train on 2% sparse observations rather than the full grid | Φ-ROM · N-S, test ICs, interpolation | RNMSE: 0.170 -> 0.189 | Hurt |
| Train on 2% sparse observations rather than the full grid | Φ-ROM · KdV, test ICs, temporal extrapolation | RNMSE: 0.486 -> 0.567 | Hurt |
Surprising findings
- DINo achieves lower error than Φ-ROM on training initial conditions and in-distribution LBM parameters, showing Φ-ROM trades in-distribution accuracy for generalization.
- PINN-ROM and AD-CROM fail grossly on the nonlinear Burgers' equation, with extrapolation RNMSE of 0.348 and 0.196.
Limitations
- Requires access to differentiable PDE solvers, limiting applicability.
- Increased training cost compared with purely data-driven methods; large-scale 3D problems not addressed.
- Limited to PDEs with first-order time derivatives; sparse-data inference on KdV performs poorly at 2% observation.
Experimental setup
Datasets: Diffusion (FD solver, 42x42 grid) Burgers' (FD solver, 256 grid) Navier-Stokes decaying turbulence (JAX-CFD, 64x64) KdV (Exponax, 64x64) LBM flow over cylinder (XLB, 180x82)
Benchmarks: Diffusion Burgers' N-S KdV LBM
Metrics: RNMSE
Baselines: DINo PINN-ROM FD-CROM AD-CROM downsampled AD-CROM
Builds on: DINo CROM PINN-ROM Neural ODE Apebench
Closest papers by meaning
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.88)
- CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs (NeurIPS 2025 · similarity 0.88)
- Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators (NeurIPS 2026 · similarity 0.88)
- Learning Reduced Fluid Dynamics (AAAI 2024 · similarity 0.88)
- Text2PDE: Latent Diffusion Models for Accessible Physics Simulation (ICLR 2025 · similarity 0.88)
- AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields (NeurIPS 2024 · similarity 0.88)
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