DynAMO: Multi-agent reinforcement learning for dynamic anticipatory mesh optimization with applications to hyperbolic conservation laws

TL;DR

DynAMO addresses the inefficiency of traditional adaptive mesh refinement for time-dependent hyperbolic PDEs by learning anticipatory refinement policies with multi-agent reinforcement learning. It formulates dynamic mesh optimization as a Dec-POMDP where element-level agents observe local, non-dimensional features and select refinement actions, trained via Independent PPO with a shared network. The approach demonstrates superior efficiency and lower error compared with threshold-based AMR across linear advection and nonlinear Euler equations, for both h- and p-refinement, and shows strong generalization to unseen problems, mesh resolutions, and longer remesh intervals. This RL-guided anticipatory refinement reduces the need for frequent remeshing and enables longer intervals between adaptations, offering practical performance benefits for large-scale, time-dependent simulations in engineering and science.

Abstract

We introduce DynAMO, a reinforcement learning paradigm for Dynamic Anticipatory Mesh Optimization. Adaptive mesh refinement is an effective tool for optimizing computational cost and solution accuracy in numerical methods for partial differential equations. However, traditional adaptive mesh refinement approaches for time-dependent problems typically rely only on instantaneous error indicators to guide adaptivity. As a result, standard strategies often require frequent remeshing to maintain accuracy. In the DynAMO approach, multi-agent reinforcement learning is used to discover new local refinement policies that can anticipate and respond to future solution states by producing meshes that deliver more accurate solutions for longer time intervals. By applying DynAMO to discontinuous Galerkin methods for the linear advection and compressible Euler equations in two dimensions, we demonstrate that this new mesh refinement paradigm can outperform conventional threshold-based strategies while also generalizing to different mesh sizes, remeshing and simulation times, and initial conditions.

Figures (37)

• Figure 1: Schematic of one-level $h$-refinement (left) and $p$-refinement on a two-dimensional quadrilateral discontinuous Galerkin finite element mesh with equal number of quadrature nodes and basis functions. Blue circles represent volume quadrature nodes, red crosses represent surface quadrature nodes.
• Figure 1: Comparison of the mean efficiency, normalized error, and normalized cost for $p$-refinement on the advection equation with DynAMO and the threshold policy for the advecting rings over 100 in-distribution runs using uniform random initial conditions. Standard deviation is shown in parentheses.
• Figure 2: Schematic for an example $h$-refinement observation setup with an initial mesh of $N = 8^2$ agents/elements, showing the mesh and example displacement vector (left), the observation window of an agent on a refined mesh (center), and the observation channels of an agent with 5 observable quantities (right).
• Figure 2: Comparison of the mean efficiency for $p$-refinement on the advection equation with DynAMO and the threshold policy for the advecting rings over 100 out-of-distribution runs using uniform random initial conditions with finer mesh resolution, different advecting shapes, and longer simulation time. Standard deviation shown in parentheses. In-distribution results from \ref{['tab:advection_pref_indistribution']} shown for comparison.
• Figure 3: Schematic for an example raw error distribution (left) transformed to a normalized error distribution (right) as a function of the hyperparameters $\alpha$ and $\beta$. Red region denotes normalized error values for which the policy penalizes de-refinement, blue region denotes normalized error values for which the policy penalizes refinement.

Theorems & Definitions (1)

• Remark : Aggregation for $h$-refinement