[Paper] Differentiable Physics-Neural Models enable Learning of Non-Markovian Closures for Accelerated Coarse-Grained Physics Simulations

Source: arXiv - 2511.21369v1

Overview

The paper introduces a hybrid physics‑neural framework that can predict scalar transport (e.g., concentration fields) on a coarse‑grained level orders of magnitude faster than a full 3‑D simulation—cutting run times from hours to under a minute. By making the whole pipeline differentiable, the authors jointly learn both the physical parameters (orthotropic diffusivity) and a non‑Markovian neural closure that captures the effects of unresolved sub‑grid dynamics, delivering stable long‑term forecasts with very little training data.

Key Contributions

• End‑to‑end differentiable surrogate that couples a traditional PDE solver with a recurrent neural network (RNN) closure, enabling joint optimization of physics parameters and data‑driven corrections.
• Non‑Markovian closure model that retains memory of past states, allowing the surrogate to emulate history‑dependent coarse‑grained effects that standard Markovian closures miss.
• Data‑efficient training: the model reaches high fidelity with only 26 simulation snapshots, demonstrating strong generalization from a tiny dataset.
• Speedup of > 10⁴×: the surrogate reproduces plane‑level concentration metrics in < 1 min versus several hours for the full 3‑D finite‑volume simulation.
• Robust out‑of‑distribution performance: when tested on a scenario with a moving source (unseen during training), the model attains a Spearman correlation of 0.96 at the final time step.

Methodology

1. Coarse‑grained physics backbone – A reduced 2‑D diffusion equation with an orthotropic diffusivity tensor (different diffusivities along orthogonal axes) captures the dominant transport physics while discarding the full 3‑D geometry.
2. Neural closure via RNN – A recurrent neural network (e.g., GRU/LSTM) receives the coarse‑grained state and outputs a corrective term added to the PDE’s right‑hand side. The recurrent nature makes the closure non‑Markovian.
3. Differentiable integration – The PDE solver is implemented with an automatic‑differentiation‑friendly discretization (e.g., finite differences using PyTorch/TensorFlow), allowing gradients to flow through both the physics solver and the neural closure during training.
4. Joint loss and optimization – The loss combines a data‑misfit term (difference between surrogate and high‑fidelity simulation at observed planes) and regularization on the diffusivity tensor. Stochastic gradient descent updates both the diffusivity parameters and the RNN weights simultaneously.
5. Training regime – Only a handful of full‑simulation snapshots are needed; the model is trained on these and then rolled out autonomously for long‑time predictions.

Results & Findings

• The surrogate reproduces the spatial distribution of concentrations on the target plane with negligible bias.
• Memory in the RNN prevents drift that typically plagues Markovian closures during long rollouts.
• Even when the source location changes (a distribution shift), the model retains high correlation, indicating strong generalization.

Practical Implications

• Rapid prototyping – Engineers can explore “what‑if” scenarios (e.g., different source locations, boundary conditions) in minutes rather than hours, accelerating design cycles for environmental modeling, chemical reactors, or HVAC systems.
• Edge deployment – Because the surrogate runs on modest GPU/CPU resources, it can be embedded in real‑time monitoring dashboards or digital twins that need near‑instant predictions.
• Data‑efficient modeling – Organizations with limited high‑fidelity simulation budgets can still build accurate surrogates, lowering computational cost and carbon footprint.
• Hybrid workflow integration – The differentiable physics‑neural pipeline fits naturally into existing ML ecosystems (PyTorch, JAX), enabling seamless integration with other data‑driven components such as sensor fusion or reinforcement learning controllers.

Limitations & Future Work

• Domain specificity – The current formulation assumes scalar diffusion in a relatively simple geometry; extending to multi‑physics (e.g., coupled momentum‑energy equations) will require more sophisticated closures.
• Scalability of the physics solver – While the surrogate is fast, the underlying differentiable PDE discretization still limits resolution; adaptive meshing or higher‑order schemes could improve fidelity.
• Interpretability of the neural closure – The RNN acts as a black box; future work could explore physics‑informed architectures or symbolic regression to extract interpretable correction terms.
• Robustness to noisy data – The study uses clean simulation outputs; assessing performance with noisy experimental measurements is an open avenue.

Bottom line: By marrying differentiable PDE solvers with memory‑rich neural closures, this work delivers a fast, accurate, and data‑lean surrogate for coarse‑grained transport problems—opening the door for real‑time, AI‑augmented physics simulations in industry.