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Metamizer: A Versatile Neural Optimizer for Fast and Accurate Physics Simulations
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Metamizer trains a scale-invariant U-Net neural optimizer via meta-learning on physics-based residual losses (no ground-truth data) to iteratively solve linear and nonlinear PDEs and cloth simulations, generalizing to unseen PDEs.
Problem
Physics simulations require solving PDEs numerically, which is computationally expensive; existing neural surrogates are tailored to specific PDEs and do not generalize across PDEs without retraining, and gradient-descent optimizers need tuning and many iterations.
Approach
A scale-invariant neural optimizer (U-Net) that takes current and previous gradients and update step, normalizes them, and predicts scaled update steps to minimize a physics-based residual loss; trained in a cyclic meta-learning loop on a training pool of randomized PDE instances without any training data.
Key results
- On the Laplace equation (100x100 grid), Metamizer reaches an MSR of 6.7e-33 after 0.08 s, competitive with CG, MINRES and GMRES and far ahead of gradient-based PyTorch optimizers.
- Advection-diffusion at 100 iterations per time step reaches MSR 2.2e-33; Navier-Stokes L_p reaches 2.4e-16 at 100 iterations per time step, orders of magnitude more accurate than prior neural approaches at 20-100 iterations.
- On a 400x400 Laplace grid Metamizer converges about 2x faster than CuPy CG/MINRES on GPU and about 10x faster than CPU CG, but AMGX is up to 10x faster than Metamizer on GPU.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase iterations per time step from 1 to 100 | Metamizer · Advection-Diffusion | mean squared residual: $7.9 \times 10^{-5}$ -> $2.2 \times 10^{-33}$ | Helped |
| Increase iterations per time step from 1 to 100 | Metamizer · Wave Equation | mean squared residual: $4.4 \times 10^{24}$ -> $3.5\times 10^{-31}$ | Helped |
| Increase iterations per time step from 1 to 100 | Metamizer · Navier-Stokes Lp | mean squared residual: $7.4\times 10^{-6}$ -> $2.4\times 10^{-16}$ | Mixed |
| Increase iterations per time step from 1 to 100 | Metamizer · Navier-Stokes Ld | mean squared residual: $1.7\times 10^{-7}$ -> $1.9\times 10^{-17}$ | Mixed |
| Increase iterations per time step from 1 to 100 | Metamizer · 2D coupled Burgers | mean squared residual: $4.1 \times 10^{-3}$ -> $5.0\times 10^{-32}$ | Helped |
| Use Metamizer instead of conjugate gradients or MINRES | Laplace equation on a $400 \times 400$ grid · Laplace equation | convergence speed: about $2\times$ | Helped |
| Use Metamizer instead of CPU-based solvers | Laplace equation on a $400 \times 400$ grid · Laplace equation | convergence speed: $10 \times$ | Helped |
| Use Nvidia AMGX instead of Metamizer | Laplace equation on a $400 \times 400$ grid · Laplace equation | runtime: up to 10 times | Hurt |
| Add a Jacobi preconditioner | CG, GMRES, and MINRES for the Laplace equation · Laplace equation | convergence speed | Hurt |
Surprising findings
- Using one iteration per time step yields lower residuals for Navier-Stokes than five iterations, because one iteration gives an incorrect steady-state wake solution.
- Naively applying the finite-difference Laplacian gives very small gradients; detaching neighbor gradients was needed to make progress.
Limitations
- Trained for a single grid size; generalization across grid sizes not shown.
- Only Dirichlet boundary conditions considered; no Neumann/Robin or self-collisions for cloth.
- Slower than highly optimized AMG solvers (AMGX up to 10x faster on GPU).
Experimental setup
Benchmarks: Laplace equation advection-diffusion equation wave equation incompressible Navier-Stokes 2D coupled Burgers equation Poisson equation cloth simulation
Metrics: mean squared residual runtime to convergence iterations per second mean squared error vs analytical ground truth
Baselines: SGD Adam AdamW RMSprop Adagrad Adadelta conjugate gradients (CuPy) MINRES GMRES LSMR AMGX pyAMG
Builds on: Learning to Optimize (L2O) U-Net Adam gradient normalization
Closest papers by meaning
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.88)
- Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.87)
- Text2PDE: Latent Diffusion Models for Accessible Physics Simulation (ICLR 2025 · similarity 0.86)
- Mosaic: A Benchmark Suite for Differentiable Physics Solvers (NeurIPS 2026 · similarity 0.86)
- MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation (ICML 2025 · similarity 0.86)
- PRDP: Progressively Refined Differentiable Physics (ICLR 2025 · similarity 0.86)
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