Abstract Simulating physical systems is essential in engineering, but analytical solutions are limited to straightforward problems. Consequently, numerical methods like the Finite Element Method (FEM) are widely used. However, the FEM becomes computationally expensive as problem complexity and accuracy demands increase. Adaptive Mesh Refinement (AMR) improves the FEM by dynamically placing mesh elements on the domain, balancing computational speed and accuracy. Classical AMR depends on heuristics or expensive error estimators, which may lead to suboptimal performance for complex simulations. While AMR methods based on machine learning are promising, they currently only scale to simple problems. In this work, we formulate AMR as a system of collaborating, homogeneous agents that iteratively split into multiple new agents. This agent-wise perspective enables a spatial reward formulation focused on reducing the maximum mesh element error. Our approach, Adaptive Swarm Mesh Refinement++ (ASMR++), offers efficient, stable optimization and generates highly adaptive meshes at user-defined resolution at inference time. Extensive experiments demonstrate that ASMR++ outperforms heuristic approaches and learned baselines, matching the performance of expensive error-based oracle AMR strategies. ASMR++ additionally generalizes to different domains during inference, and produces meshes that simulate up to 2 orders of magnitude faster than uniform refinements in more demanding settings.

Accurately simulating physics is crucial across scientific domains, with applications spanning from robotics to materials science. While traditional mesh-based simulations are precise, they are often computationally expensive and require knowledge of physical parameters, such as material properties. In contrast, data-driven approaches like Graph Network Simulators (GNSs) offer faster inference but suffer from two key limitations: Firstly, they must be retrained from scratch for even minor variations in physical parameters, and secondly they require labor-intensive data collection for each new parameter setting. This is inefficient, as simulations with varying parameters often share a common underlying latent structure. In this work, we address these challenges by learning this shared structure through meta-learning, enabling fast adaptation to new physical parameters without retraining. To this end, we propose a novel architecture that generates a latent representation by encoding graph trajectories using conditional neural processes (CNPs). To mitigate error accumulation over time, we combine CNPs with a novel neural operator architecture. We validate our approach, Meta Neural Graph Operator (MaNGO), on several dynamics prediction tasks with varying material properties, demonstrating superior performance over existing GNS methods. Notably, MaNGO achieves accuracy on unseen material properties close to that of an oracle model.

The cost and accuracy of simulating complex physical systems using the Finite Element Method (FEM) scales with the resolution of the underlying mesh. Adaptive meshes improve computational efficiency by refining resolution in critical regions, but typically require task-specific heuristics or cumbersome manual design by a human expert. We propose Adaptive Meshing By Expert Reconstruction (AMBER), a supervised learning approach to mesh adaptation. Starting from a coarse mesh, AMBER iteratively predicts the sizing field, i.e., a function mapping from the geometry to the local element size of the target mesh, and uses this prediction to produce a new intermediate mesh using an out-of-the-box mesh generator. This process is enabled through a hierarchical graph neural network, and relies on data augmentation by automatically projecting expert labels onto AMBER-generated data during training. We evaluate AMBER on 2D and 3D datasets, including classical physics problems, mechanical components, and real-world industrial designs with human expert meshes. AMBER generalizes to unseen geometries and consistently outperforms multiple recent baselines, including ones using Graph and Convolutional Neural Networks, and Reinforcement Learning-based approaches.

The Finite Element Method, an important technique in engineering, is aided by Adaptive Mesh Refinement (AMR), which dynamically refines mesh regions to allow for a favorable trade-off between computational speed and simulation accuracy. Classical methods for AMR depend on task-specific heuristics or expensive error estimators, hindering their use for complex simulations. Recent learned AMR methods tackle these problems, but so far scale only to simple toy examples. We formulate AMR as a novel Adaptive Swarm Markov Decision Process in which a mesh is modeled as a system of simple collaborating agents that may split into multiple new agents. This framework allows for a spatial reward formulation that simplifies the credit assignment problem, which we combine with Message Passing Networks to propagate information between neighboring mesh elements. We experimentally validate the effectiveness of our approach, Adaptive Swarm Mesh Refinement (ASMR), showing that it learns reliable, scalable, and efficient refinement strategies on a set of challenging problems. Our approach significantly speeds up computation, achieving up to 30-fold improvement compared to uniform refinements in complex simulations. Additionally, we outperform learned baselines and achieve a refinement quality that is on par with a traditional error-based AMR strategy without expensive oracle information about the error signal.