M2N: Mesh Movement Networks for PDE Solvers

Song, Weijie; Zhang, Mingrui; Wallwork, Joseph G.; Gao, Jian; Tian, Zheng; Sun, Fanglei; Piggott, Matthew D.; Chen, Junqing; Shi, Zuoqiang; Chen, Xiang; Wang, Jun

doi:10.48550/arxiv.2204.11188

Cited by 2 publications

(4 citation statements)

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“…[10] proposed a local single-agent RL approach whereby the agent makes a decision for one randomly-selected element at each step. At training time, the global solution is updated every time a Other work at the intersection of FEM and deep learning include reinforcement learning for generating a fixed (nonadaptive) mesh [26], unsupervised clustering for marking and p-refinement [33], and supervised learning for target resolution prediction [23], error estimation [37], and mesh movement [30].…”

Section: Related Workmentioning

confidence: 99%

Multi-Agent Reinforcement Learning for Adaptive Mesh Refinement

Yang¹,

Mittal²,

Dzanic³

et al. 2022

Preprint

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Adaptive mesh refinement (AMR) is necessary for efficient finite element simulations of complex physical phenomenon, as it allocates limited computational budget based on the need for higher or lower resolution, which varies over space and time. We present a novel formulation of AMR as a fully-cooperative Markov game, in which each element is an independent agent who makes refinement and de-refinement choices based on local information. We design a novel deep multi-agent reinforcement learning (MARL) algorithm called Value Decomposition Graph Network (VDGN), which solves the two core challenges that AMR poses for MARL: posthumous credit assignment due to agent creation and deletion, and unstructured observations due to the diversity of mesh geometries. For the first time, we show that MARL enables anticipatory refinement of regions that will encounter complex features at future times, thereby unlocking entirely new regions of the error-cost objective landscape that are inaccessible by traditional methods based on local error estimators. Comprehensive experiments show that VDGN policies significantly outperform error threshold-based policies in global error and cost metrics. We show that learned policies generalize to test problems with physical features, mesh geometries, and longer simulation times that were not seen in training. We also extend VDGN with multi-objective optimization capabilities to find the Pareto front of the tradeoff between cost and error.

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Section: Related Workmentioning

confidence: 99%

Multi-Agent Reinforcement Learning for Adaptive Mesh Refinement

Yang¹,

Mittal²,

Dzanic³

et al. 2022

Preprint

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show abstract

“…These include the test function choice [7], forward solve [8], adjoint solve [9], derivative recovery procedure [10], error estimation [11], metric/monitor function/sizing field construction step [12,13,14], and the entire mesh adaptation loop [6,15,16].…”

Section: Accelerating Mesh Adaptation Using Neural Networkmentioning

confidence: 99%

“…They then solve the full PDE using conventional methods on the final mesh, so that physics are accurately represented. The authors of [16] go further and propose an end-to-end r-adaptation method, whereby the solution on the current mesh is mapped directly to the adapted mesh, removing the need to solve any MMPDEs. This can be particularly beneficial for highly nonlinear MMPDEs such as those of Monge-Ampère type, which can be computationally expensive to solve.…”

Section: Accelerating Mesh Adaptation Using Neural Networkmentioning

confidence: 99%

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E2N: Error Estimation Networks for Goal-Oriented Mesh Adaptation

Wallwork¹,

Lu²,

Zhang³

et al. 2022

Preprint

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Given a partial differential equation (PDE), goal-oriented error estimation allows us to understand how errors in a diagnostic quantity of interest (QoI), or goal, occur and accumulate in a numerical approximation, for example using the finite element method. By decomposing the error estimates into contributions from individual elements, it is possible to formulate adaptation methods, which modify the mesh with the objective of minimising the resulting QoI error. However, the standard error estimate formulation involves the true adjoint solution, which is unknown in practice. As such, it is common practice to approximate it with an 'enriched' approximation (e.g. in a higher order space or on a refined mesh). Doing so generally results in a significant increase in computational cost, which can be a bottleneck compromising the competitiveness of (goal-oriented) adaptive simulations. The central idea of this paper is to develop a "data-driven" goal-oriented mesh adaptation approach through the selective replacement of the expensive error estimation step with an appropriately configured and trained neural network. In doing so, the error estimator may be obtained without even constructing the enriched spaces. An element-byelement construction is employed here, whereby local values of various parameters related to the mesh geometry and underlying problem physics are taken as inputs, and the corresponding contribution to the error estimator is taken as output. We demonstrate that this approach is able to obtain the same accuracy with a reduced computational cost, for adaptive mesh test cases related to flow around tidal turbines, which interact via their downstream wakes, and where the overall power output of the farm is taken as the QoI. Moreover, we demonstrate that the element-byelement approach implies reasonably low training costs.

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M2N: Mesh Movement Networks for PDE Solvers

Cited by 2 publications

References 0 publications

Multi-Agent Reinforcement Learning for Adaptive Mesh Refinement

Multi-Agent Reinforcement Learning for Adaptive Mesh Refinement

E2N: Error Estimation Networks for Goal-Oriented Mesh Adaptation

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