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Adaptive hybrid coupling with operator inference, the overlapping Schwarz alternating method and reinforcement learning

Trishit Mondal, Irina Tezaur, Anthony Gruber

arXiv:2609.17837v1cs.LGcs.AImath-phmath.MP

TL;DR

Fixed FOM/ROM assignments cannot follow transient features as they move across subdomains. The paper trains RL policies to adapt model assignments online within O-SAM, and reports improved comparisons with fixed assignments on both benchmarks. The learned policies assign FOMs to regions containing propagating features and ROMs elsewhere.

  • Problem

    Fixed model assignments cannot account for changing dynamics or traveling features, risking propagated inaccuracies when such features enter ROM-assigned subdomains.

    Source: §1 · p. 2

  • Method

    RL learns online subdomain model-selection policies within O-SAM using rewards that balance computational cost, solution error, and switching penalties.

    Source: §1 · p. 3, §3.3 · p. 12

  • Results

    Across both benchmarks, learned policies improved on the most accurate fixed assignments containing at least one ROM; elastic-wave errors were more than an order of magnitude smaller.

    Source: §5 · p. 28

  • Takeaways & Limitations

    The policies adapt FOM/ROM assignments as fronts and waves propagate, placing FOMs in subdomains containing the wave and ROMs elsewhere.

    Source: §5 · p. 28

    The ROMs are pretrained and are not updated online using newly available FOM information.

    Source: §1 · p. 3

Abstract

from arXiv · show

Hybrid domain decomposition methods provide a flexible framework for coupling full order models (FOMs) and reduced order models (ROMs), but typically assume the model assigned to each subdomain is fixed throughout a simulation. This is limiting for transient problems in which localized features propagate through the domain and the regions requiring high-fidelity resolution change over time. We introduce a reinforcement learning (RL)-based approach for online adaptation of FOM-ROM models coupled via the overlapping Schwarz alternating method (O-SAM), an iterative domain decomposition method that solves subdomain-local problems while exchanging solution information through transmission boundary conditions on overlapping interfaces. Deep Q-networks (DQNs) are trained offline to select among subdomain-local FOMs and pre-trained Operator Inference (OpInf) ROMs using a reward balancing accuracy, cost, and model-switching frequency. Once trained, the policies are deployed predictively on problem instances not seen during training, without requiring a reference FOM solution. We demonstrate the approach on two examples: a 1D advection-diffusion problem with a moving front, and a 3D linear elastic wave propagation problem implemented in the Norma.jl solid mechanics code. For the advection-diffusion benchmark, the learned policy dynamically allocates high-fidelity resolution as the front propagates and outperforms static FOM/ROM assignments; letting the agent also adapt the domain decomposition provides no further benefit. For the elastic wave benchmark, learned policies for two and three subdomain decompositions track the propagating wave by assigning FOMs to subdomains containing the wave and ROMs elsewhere, as expected. Our results demonstrate the potential of RL to enable predictive online adaptation of model fidelity within Schwarz-based hybrid simulations.

1. Introduction.

Hybrid domain decomposition couples local models, but fixed subdomain assignments cannot follow changing or traveling features. This work uses reinforcement learning to adapt FOM/ROM assignments online within O-SAM.

1. Introduction.

  • Fixed subdomain model assignments cannot follow changing dynamics or traveling features, which can introduce and propagate inaccuracies when a feature enters a ROM-assigned region.The paper identifies this as the primary limitation of the prior fixed-assignment workflow.Source: §1 · p. 2
  • RL–O-SAM learns when and where to assign subdomain-local FOMs or OpInf ROMs, balancing solution error, computational cost, and unnecessary switching.Policies are trained on problem instances and deployed predictively on instances not seen during training.Source: §1 · p. 3
  • The ROMs are pretrained and are not updated online using newly available FOM information.Online OpInf ROM updates are identified as future work.Source: §1 · p. 3

2.1. Operator Inference (OpInf).

  • OpInf is a projection-inspired, non-intrusive model reduction approach that does not require access to the high-fidelity simulation code.Its workflow includes constructing a reduced basis from snapshots, learning reduced operators, and deploying the learned ROM online.Source: §2.1 · pp. 3–4

2.2. O-SAM applied to Operator Inference.

  • O-SAM replaces a global problem with sequential subdomain-local solves that exchange transmission boundary conditions across interfaces.For overlapping subdomains with nonempty overlap, Dirichlet transmission conditions guarantee convergence when the underlying problem is well-posed.Source: §2.2 · p. 5
  • O-SAM permits plug-and-play coupling of different models and meshes, including conventional and data-driven models, across subdomains.The paper uses this flexibility to couple subdomain-local FOMs and OpInf ROMs, and proposes adapting both model assignments and the decomposition over time.Source: §2.2 · pp. 5, 7

3. Reinforcement learning for online adaptation of O-SAM.

The framework uses reinforcement learning to adapt FOM/ROM assignments within O-SAM as a simulation evolves, while leaving the Schwarz iteration unchanged. A DQN is trained offline and then deployed to select configurations from observed states.

3.1. Reinforcement learning.

  • Reinforcement learning frames simulation adaptation as sequential decisions: the agent observes a state, chooses an action, and receives a reward that balances accuracy and cost.DQN is suited to the high-dimensional solution states and discrete set of model assignments considered here.Source: §3.1 · pp. 7–8, §3.2 · p. 8

3.2. Deep Q-networks (DQNs).

  • DQN approximates action values with a neural network, enabling estimates across high-dimensional states; replay-buffer sampling and a separate target network support training.The online network selects the next action, while the target network evaluates it, reducing systematic overestimation of action values.Source: §3.2 · p. 9

3.3. RL-based online adaptation of O-SAM.

  • Fixed-DD actions enumerate FOM/ROM assignments across prescribed subdomains, while adaptive-DD actions also select among candidate decompositions.For adaptive-DD, the action space contains NDD × 2^nd configurations.Source: §3.3 · p. 11
  • Offline training rewards balance computational cost, error relative to a high-fidelity reference, and changes in the selected configuration.The policy is trained from transitions collected across problem instances with differing initial conditions, boundary conditions, and/or physical parameters.Source: §3.3 · p. 12
  • The trained policy selects subdomain models, and optionally the domain decomposition, once per decision window while O-SAM advances the coupled solution.At deployment, the policy acts from the observed state without a reference solution or reward evaluation.Source: §3.2 · p. 10, §3.3 · p. 12

4. Numerical results.

The numerical results examine how learned policies adapt FOM/ROM assignments in advection–diffusion and elastic-wave benchmarks. They report accuracy, cost, and timing outcomes across fixed and adaptive assignment strategies.

4.1.1. Fixed-DD case.

4.1.1.1. Effect of the switching penalty.

4.1.2. Adaptive-DD case.

4.2. Clamped linear elastic wave propagation benchmark.

  • The elastic-wave benchmark uses the 3D Norma.jl code, with transverse-displacement Dirichlet conditions restricting dynamics to the axial direction.Source: §4.1.2 · p. 22, §4.2 · p. 22
  • Policies were tested on 10 held-out pulse location/width combinations among 21 cases; ROMs were trained on all 21, while domain decompositions stayed fixed.The fixed-decomposition choice followed the advection-diffusion finding that adaptive decomposition did not improve the accuracy-cost tradeoff, and because dynamic decomposition was unsupported in Norma.jl. The reward error compared assembled displacement, velocity, and acceleration fields with the all-FOM O-SAM reference.Source: §4.2 · pp. 23–24

4.3. Two subdomain study.

  • In the two-subdomain test, the policy moved the FOM assignment with the propagating and reflected wave, using ROMs where the solution was comparatively smooth.During the reflection, it temporarily assigned FOMs to both subdomains.Source: §4.3 · p. 25, Figure 4.10 · p. 25
  • The two-subdomain policy achieved relative L2 error 1.59 × 10−2 versus 2.43 × 10−1 for the best ROM-containing fixed assignment (RF), with wall time 49.77 s versus 52.91 s for all-FOM (FF).The all-ROM (RR) assignment had the lowest wall time, 33.23 s, but relative L2 error 2.69 × 10−1.Source: §4.3 · p. 25, Table 4.4 · p. 26

4.4. Three subdomain study.

  • In the three-subdomain test, the policy shifted FOMs among subdomains as the wave traveled and reflected, assigning ROMs to regions outside the wave.This behavior persisted with eight possible FOM/ROM assignments.Source: Figure 4.11 · p. 26, §4.4 · p. 27
  • The three-subdomain policy achieved relative L2 error 3.90 × 10−3 versus 1.54×10−1 for the best ROM-containing fixed assignment (FFR), with wall time 54.04 s versus 62.63 s for all-FOM (FFF).The all-ROM (RRR) assignment had the lowest wall time, 42.10 s, but relative L2 error 2.94 × 10−1.Source: §4.4 · p. 27, Table 4.5 · p. 27

5. Conclusions.

The learned policies adapted FOM/ROM assignments to moving fronts and waves, improving the accuracy-cost tradeoff over fixed assignments in the tested benchmarks. The study identifies fixed-decomposition adaptation as effective for the advection-diffusion case and outlines online ROM adaptation and more advanced RL as future directions.

5. Conclusions.

  • Offline-trained policies select subdomain-local FOMs or pre-trained OpInf ROMs using a reward balancing accuracy, computational cost, and switching frequency, then run predictively without a reference FOM solution.The approach was evaluated on a 1D moving-front advection-diffusion problem and a 3D linear elastic wave problem in Norma.jl, with predictive policy evaluation on unseen instances.Source: §5 · p. 28
  • For advection-diffusion, the fixed-domain policy achieved relative L2 error 8.02 × 10−4, about five times smaller than the most accurate fixed assignment containing a ROM, while reducing wall time versus all-FOM coupling.Adapting the domain decomposition did not improve the accuracy-cost tradeoff; its policy had higher error and wall time despite slightly fewer Schwarz iterations.Source: §5 · p. 28
  • For elastic waves, learned policies assigned FOMs to subdomains containing the propagating wave and ROMs elsewhere, achieving errors more than an order of magnitude below the most accurate fixed assignments containing a ROM while reducing wall time versus all-FOM assignments.The policies generalized to held-out pulse locations and widths.Source: §5 · p. 28
  • Future work includes applying the workflow to more complex nonlinear solid mechanics problems and enabling OpInf ROMs to adapt online using FOM information generated during hybrid simulation.The current OpInf ROMs are constructed offline and remain fixed during deployment; the authors also identify more advanced RL methods for larger or more structured action spaces as a direction.Source: §5 · pp. 28–29

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