# Generalized Neural Closure Models with Interpretability

arXiv: https://arxiv.org/abs/2301.06198 · HTML: https://mlsanity.com/paper/2301.06198/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 79 references that are in the map.
- First posted on arXiv 15 Jan 2023.

## Builds on (most cited first, 15 of 15)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs](https://mlsanity.com/paper/1902.10298/) (2019)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)](https://mlsanity.com/paper/1911.09145/) (2019)
- [Neural Closure Models for Dynamical Systems](https://mlsanity.com/paper/2012.13869/) (2020)
- [Data-driven Discovery of Closure Models](https://mlsanity.com/paper/1803.09318/) (2018)
- [Automatic differentiation in ML: Where we are and where we should be going](https://mlsanity.com/paper/1810.11530/) (2018)
- [Tangent: Automatic Differentiation Using Source Code Transformation in Python](https://mlsanity.com/paper/1711.02712/) (2017)
- [Sparse Regression and Adaptive Feature Generation for the Discovery of Dynamical Systems](https://mlsanity.com/paper/1902.02719/) (2019)
- [Batch Size Influence on Performance of Graphic and Tensor Processing Units during Training and Inference Phases](https://mlsanity.com/paper/1812.11731/) (2018)
- [A Deep Learning Approach for Predicting Spatiotemporal Dynamics From Sparsely Observed Data](https://mlsanity.com/paper/2011.14965/) (2020)
- [A memory-efficient neural ODE framework based on high-level adjoint differentiation](https://mlsanity.com/paper/2206.01298/) (2022)
- [Bayesian Learning of Coupled Biogeochemical-Physical Models](https://mlsanity.com/paper/2211.06714/) (2022)

## Built on by (most cited first, 8 of 8)

- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [Evaluation of Deep Neural Operator Models toward Ocean Forecasting](https://mlsanity.com/paper/2308.11814/) (2023)
- [Learning Stochastic Multiscale Models](https://mlsanity.com/paper/2506.22655/) (2025)
- [Guided Unconditional and Conditional Generative Models for Super-Resolution and Inference of Quasi-Geostrophic Turbulence](https://mlsanity.com/paper/2507.00719/) (2025)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [Differentiable Physics-Neural Models enable Learning of Non-Markovian Closures for Accelerated Coarse-Grained Physics Simulations](https://mlsanity.com/paper/2511.21369/) (2025)
- [Synergizing Transport-Based Generative Models and Latent Geometry for Stochastic Closure Modeling](https://mlsanity.com/paper/2602.17089/) (2026)
- [Smoothness Errors in Dynamics Models and How to Avoid Them](https://mlsanity.com/paper/2602.05352/) (2026)

## Abstract (arXiv)

Improving the predictive capability and computational cost of dynamical models is often at the heart of augmenting computational physics with machine learning (ML). However, most learning results are limited in interpretability and generalization over different computational grid resolutions, initial and boundary conditions, domain geometries, and physical or problem-specific parameters. In the present study, we simultaneously address all these challenges by developing the novel and versatile methodology of unified neural partial delay differential equations. We augment existing/low-fidelity dynamical models directly in their partial differential equation (PDE) forms with both Markovian and non-Markovian neural network (NN) closure parameterizations. The melding of the existing models with NNs in the continuous spatiotemporal space followed by numerical discretization automatically allows for the desired generalizability. The Markovian term is designed to enable extraction of its analytical form and thus provides interpretability. The non-Markovian terms allow accounting for inherently missing time delays needed to represent the real world. We obtain adjoint PDEs in the continuous form, thus enabling direct implementation across differentiable and non-differentiable computational physics codes, different ML frameworks, and treatment of nonuniformly-spaced spatiotemporal training data. We demonstrate the new generalized neural closure models (gnCMs) framework using four sets of experiments based on advecting nonlinear waves, shocks, and ocean acidification models. Our learned gnCMs discover missing physics, find leading numerical error terms, discriminate among candidate functional forms in an interpretable fashion, achieve generalization, and compensate for the lack of complexity in simpler models. Finally, we analyze the computational advantages of our new framework.