Adaptive recurrent flow map operator learning for reaction diffusion dynamics

TL;DR

The paper tackles the problem of learning stable, long-horizon flow-map operators for reaction–diffusion dynamics using purely data-driven methods. It introduces DDOL-ART, an adaptive recurrent training strategy that couples rollout dynamics with lightweight validation milestones to curb error accumulation and drift, reducing training cost while preserving robustness. Through extensive experiments on FitzHugh–Nagumo, Gray–Scott, and Lambda–Omega systems, DDOL-ART demonstrates competitive long-horizon accuracy and zero-shot generalization to strong morphology shifts, outperforming physics-based residual approaches in several regimes and delivering 3.2×–3.6× training speedups. The findings suggest that feedback-controlled recurrent supervision can be a broadly transferable approach for stable dynamical-system forecasting when governing equations are unavailable, with potential extension to partial observations and non-periodic geometries.

Abstract

Reaction-diffusion (RD) equations underpin pattern formation across chemistry, biology, and physics, yet learning stable operators that forecast their long-term dynamics from data remains challenging. Neural-operator surrogates provide resolution-robust prediction, but autoregressive rollouts can drift due to the accumulation of error, and out-of-distribution (OOD) initial conditions often degrade accuracy. Physics-based numerical residual objectives can regularize operator learning, although they introduce additional assumptions, sensitivity to discretization and loss design, and higher training cost. Here we develop a purely data-driven operator learner with adaptive recurrent training (DDOL-ART) using a robust recurrent strategy with lightweight validation milestones that early-exit unproductive rollout segments and redirect optimization. Trained only on a single in-distribution toroidal Gaussian family over short horizons, DDOL-ART learns one-step operators that remain stable under long rollouts and generalize zero-shot to strong morphology shifts across FitzHugh-Nagumo (FN), Gray-Scott (GS), and Lambda-Omega (LO) systems. Across these benchmarks, DDOL-ART delivers a strong accuracy and cost trade-off. It is several-fold faster than a physics-based numerical-loss operator learner (NLOL) under matched settings, and it remains competitive on both in-distribution stability and OOD robustness. Training-dynamics diagnostics show that adaptivity strengthens the correlation between validation error and OOD test error performance, acting as a feedback controller that limits optimization drift. Our results indicate that feedback-controlled recurrent training of DDOL-ART generates robust flow-map surrogates without PDE residuals, while simultaneously maintaining competitiveness with NLOL at significantly reduced training costs.

Figures (8)

• Figure 1: ID performance of the operator learning models. For each benchmark FN, GS, and LO, bars show the $\Delta t$-averaged AMAE obtained by NLOL, DDOL, and DDOL-ART models with $B\in\{32,64\}$, $T\in\{1,5\}$ and $\Delta t\in\{0.005,0.01,0.02,0.05,0.1\}$. We considered $10$ random seeds to generate ID initial conditions and measured AMAE using the final test time $T=10$ for each model. The presented values are the average of the AMAE values of five different models trained using $\Delta t\in\{0.005,0.01,0.02,0.05,0.1\}$. Entries are sorted in ascending order, with the best in each subplot highlighted in green.
• Figure 2: GS training-dynamics ablation of the DDOL-ART failure threshold $n_{\mathrm{fail}}$ under fixed hyperparameters ($B=32$, $T=1$, $\Delta t=0.01$), averaged over three random seeds. (A) Batch-resolved validation (ID) and OOD-test MSE during training progress including two outer epochs and eight batches for each outer epoch. Each batch includes four ID ICs during training. (B) Cumulative training time versus OOD-test MSE.
• Figure 3: Batch-level validation MSE versus OOD-test MSE, together with Pearson correlation coefficients ($r$) and two-sided $p$-values.
• Figure 4: Out-of-distribution performance of NLOL, DDOL, and DDOL-ART on three unseen initial-condition families: multi-Gaussian, noisy Gaussian, and patch. All models are trained only on the in-distribution single-Gaussian family using the fixed ID-selected configurations from Table \ref{['tab:model_selection']}, and are evaluated by rolling out to $T_{\text{test}}=10$ over $10$ random seeds. Bars are sorted in ascending order, and the best method in each panel is highlighted in green.
• Figure 5: Time-resolved MAE up to $T_{\text{test}}=10$ for NLOL, DDOL, and DDOL-ART models. Columns show different problems FN, GS, and LO. Rows show the generalization performance of each operator learning model for ID (single Gaussian) and the OOD families: multi-Gaussian, noisy Gaussian, and patch. At each time $t$, curves report MAE averaged over ten independently sampled initial conditions of the indicated family.