# Neural Closure Models for Dynamical Systems

arXiv: https://arxiv.org/abs/2012.13869 · HTML: https://mlsanity.com/paper/2012.13869/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Dec 2020.

## Built on by (most cited first, 22 of 22)

- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
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- [A Multifidelity deep operator network approach to closure for multiscale systems](https://mlsanity.com/paper/2303.08893/) (2023)
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- [Output-weighted and relative entropy loss functions for deep learning precursors of extreme events](https://mlsanity.com/paper/2112.00825/) (2021)
- [Generalized Neural Closure Models with Interpretability](https://mlsanity.com/paper/2301.06198/) (2023)
- [A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic](https://mlsanity.com/paper/2204.06362/) (2022)
- [Comparison of neural closure models for discretised PDEs](https://mlsanity.com/paper/2210.14675/) (2022)
- [Low-dimensional Data-based Surrogate Model of a Continuum-mechanical Musculoskeletal System Based on Non-intrusive Model Order Reduction](https://mlsanity.com/paper/2302.06528/) (2023)
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- [Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling](https://mlsanity.com/paper/2205.12419/) (2022)
- [Evaluation of Deep Neural Operator Models toward Ocean Forecasting](https://mlsanity.com/paper/2308.11814/) (2023)
- [Bayesian Learning of Coupled Biogeochemical-Physical Models](https://mlsanity.com/paper/2211.06714/) (2022)
- [Guided Unconditional and Conditional Generative Models for Super-Resolution and Inference of Quasi-Geostrophic Turbulence](https://mlsanity.com/paper/2507.00719/) (2025)
- [Neural delay differential equations: learning non-Markovian closures for partially known dynamical systems](https://mlsanity.com/paper/2410.02843/) (2024)
- [Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems](https://mlsanity.com/paper/2207.12854/) (2022)
- [Automatic and Structure-Aware Sparsification of Hybrid Neural ODEs](https://mlsanity.com/paper/2505.18996/) (2025)
- [DSSMs: State Space Models with Explicit Memory via Delay Differential Equations](https://mlsanity.com/paper/2607.10244/) (2026)

## Abstract (arXiv)

Complex dynamical systems are used for predictions in many domains. Because of computational costs, models are truncated, coarsened, or aggregated. As the neglected and unresolved terms become important, the utility of model predictions diminishes. We develop a novel, versatile, and rigorous methodology to learn non-Markovian closure parameterizations for known-physics/low-fidelity models using data from high-fidelity simulations. The new "neural closure models" augment low-fidelity models with neural delay differential equations (nDDEs), motivated by the Mori-Zwanzig formulation and the inherent delays in complex dynamical systems. We demonstrate that neural closures efficiently account for truncated modes in reduced-order-models, capture the effects of subgrid-scale processes in coarse models, and augment the simplification of complex biological and physical-biogeochemical models. We find that using non-Markovian over Markovian closures improves long-term prediction accuracy and requires smaller networks. We derive adjoint equations and network architectures needed to efficiently implement the new discrete and distributed nDDEs, for any time-integration schemes and allowing nonuniformly-spaced temporal training data. The performance of discrete over distributed delays in closure models is explained using information theory, and we find an optimal amount of past information for a specified architecture. Finally, we analyze computational complexity and explain the limited additional cost due to neural closure models.