Metamizer: a versatile neural optimizer for fast and accurate physics simulations

TL;DR

Metamizer proposes a scale-invariant neural optimizer that accelerates physics-based gradient descent without requiring ground-truth data, enabling fast and highly accurate simulations across a wide range of linear and nonlinear PDEs. By predicting adaptive, scale-aware update steps via a U-Net trained in a meta-learning loop, it achieves near-machine-precision solutions for problems like Laplace, Poisson, advection-diffusion, Navier–Stokes, Burgers, and cloth dynamics, and generalizes to unseen PDEs such as the wave equation. This approach offers a single, reusable model with a tunable speed-accuracy trade-off, competitive with sparse linear solvers and capable of real-time or near-real-time physics in computer graphics and engineering contexts. The work points toward broader impacts in numerical solvers and suggests extensions to more boundary conditions, mesh/graph representations, and coupled PDE systems.

Abstract

Efficient physics simulations are essential for numerous applications, ranging from realistic cloth animations or smoke effects in video games, to analyzing pollutant dispersion in environmental sciences, to calculating vehicle drag coefficients in engineering applications. Unfortunately, analytical solutions to the underlying physical equations are rarely available, and numerical solutions require high computational resources. Latest developments in the field of physics-based Deep Learning have led to promising efficiency improvements but still suffer from limited generalization capabilities and low accuracy compared to numerical solvers. In this work, we introduce Metamizer, a novel neural optimizer that iteratively solves a wide range of physical systems with high accuracy by minimizing a physics-based loss function. To this end, our approach leverages a scale-invariant architecture that enhances gradient descent updates to accelerate convergence. Since the neural network itself acts as an optimizer, training this neural optimizer falls into the category of meta-optimization approaches. We demonstrate that Metamizer achieves unprecedented accuracy for deep learning based approaches - sometimes approaching machine precision - across multiple PDEs after training on the Laplace, advection-diffusion and incompressible Navier-Stokes equation as well as on cloth simulations. Remarkably, the model also generalizes to PDEs that were not covered during training such as the Poisson, wave and Burgers equation. Our results suggest that Metamizer could have a profound impact on future numerical solvers, paving the way for fast and accurate neural physics simulations without the need for retraining.

Figures (25)

• Figure 1: Metamizer is able to simulate various linear and non-linear physical systems. All of the depicted results were produced by the same neural model (same architecture and same weights). PDEs marked in red were not considered during training.
• Figure 2: Exemplary visualization of a 2D loss function with a steep valley. Naive gradient descent fails since it either converges very slowly (blue curve 1) or diverges (red curve 2). Ideally, we would like to adapt the step size and direction for accelerated convergence (green curve 3).
• Figure 3: a) Metamizer architecture: a scale invariant neural optimizer. b) Training Cycle.
• Figure 4: Iterative refinements by Metamizer to solve the Laplace equation. Top row: Intermediate solutions of $u$. Middle row: corresponding gradients of the physics-based loss. Bottom row: update steps computed by Metamizer. An animation of this process is presented in the supplemental video.
• Figure 5: The scale parameter $s_i$ (see Figure \ref{['fig:architecture_cycle']} a) is automatically adjusted by Metamizer to make appropriate update steps. a) stationary Laplace Equation b) time-dependent advection-diffusion equation with 10 iterations per time step.