Research map / Probabilistic and scientific modeling / Differential equations and PDEs
PRDP: Progressively Refined Differentiable Physics
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes PRDP, which starts neural network training with coarse iterative linear solves in differentiable physics and adaptively increases solver iterations when validation plateaus, cutting training cost while keeping accuracy.
Problem
Training neural networks through differentiable physics solvers is expensive because iterative linear solvers must be unrolled or differentiated through many iterations, and the required solver accuracy is unknown a priori.
Approach
Train with incompletely converged physics solvers (few iterations) and adaptively refine the number of solver iterations when validation metrics plateau, stopping at the adequate refinement level; applies to both unrolled and implicit differentiation, combining progressive-refinement and incomplete-convergence savings.
Key results
- PRDP reduces total physics solver iterations by up to 86% (heat 2D) relative to fully converged training, with 78% reduction in training time for 3D heat case.
- In the Navier-Stokes neural-hybrid emulator, PRDP reduces inner iterations by 81% and wall-clock training time by 62%.
- On the Poisson inverse problem, PRDP achieves 33% progressive-refinement savings, versus 28% for a hand-tuned linear schedule.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| adaptively refine solver iterations and stop at adequate refinement | PRDP on the Poisson inverse problem · Poisson inverse problem | inner-iteration savings: 33% | Helped |
| use PRDP instead of fully converged physics | Heat 1D emulator training · Heat 1D | solver-iteration savings: 79% | Helped |
| use PRDP instead of fully converged physics | Heat 2D emulator training · Heat 2D | solver-iteration savings: 86% | Helped |
| use PRDP instead of fully converged physics | Heat 2D emulator training · Heat 2D | training time: 24% | Helped |
| use PRDP instead of fully converged physics | Heat 3D emulator training · Heat 3D | solver-iteration savings: 81% | Helped |
| use PRDP instead of fully converged physics | Heat 3D emulator training · Heat 3D | training time: 78% | Helped |
| use PRDP instead of fully converged physics | Burgers 1D emulator training · Burgers 1D | solver-iteration savings: 59% | Helped |
| use PRDP instead of fully converged physics | Navier-Stokes neural-hybrid emulator training · Navier-Stokes neural-hybrid emulator | solver-iteration savings: 81% | Helped |
| use PRDP instead of fully converged physics | Navier-Stokes neural-hybrid emulator training · Navier-Stokes neural-hybrid emulator | training time: 62% | Helped |
| train with incompletely converged physics at K_max rather than fully converged physics | Heat 1D emulator · Heat 1D | validation accuracy: K_max = 10; K_epsilon = 25 | No clear effect |
| train with incompletely converged physics at K_max rather than fully converged physics | Heat 2D emulator · Heat 2D | solver-iteration savings: 72% | Helped |
| start training at K_0 = 4 rather than a coarser initial solver level | Burgers 1D emulator training · Burgers 1D | training convergence | Helped |
Surprising findings
- Network accuracy saturates at a physics refinement far below full solver convergence (e.g., K_max=10 vs K_epsilon=25 for 1D heat), so fully converged physics is unnecessary for training.
- Training with very coarse physics (K=1 or 2) can converge to a worse value than initialization, requiring a minimum refinement level K_min.
Limitations
- Only validated on linear solvers with sparse discretizations on structured grids; unstructured discretizations and multi-physics untested.
- Not applicable to purely explicit numerical solvers or spectral solves.
- Neural-hybrid validation requires running the fully converged solver at inference, adding cost to validation.
Experimental setup
Datasets: simulated PDE data (Poisson, heat diffusion 1D/2D/3D, Burgers, Navier-Stokes)
Benchmarks: Poisson inverse problem heat diffusion 1D/2D/3D emulation Burgers emulation Navier-Stokes neural-hybrid correction
Metrics: normalized MSE total inner solver iterations wall-clock training time reduction
Baselines: fully converged physics training (K_epsilon) manually scheduled linear inner iterations
Builds on: bilevel optimization with inexact inner solvers unrolled differentiation implicit differentiation Deep Equilibrium Models
Closest papers by meaning
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.89)
- The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.88)
- Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers (NeurIPS 2025 · similarity 0.87)
- A Physics-preserved Transfer Learning Method for Differential Equations (NeurIPS 2025 · similarity 0.87)
- Breakeven complexity: A new perspective on neural partial differential equation solvers (NeurIPS 2026 · similarity 0.87)
- Efficient Error Certification for Physics-Informed Neural Networks (ICML 2024 · similarity 0.87)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.