# Learning reduced systems via deep neural networks with memory

arXiv: https://arxiv.org/abs/2003.09451 · HTML: https://mlsanity.com/paper/2003.09451/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Mar 2020.

## Built on by (most cited first, 20 of 20)

- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)
- [Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks](https://mlsanity.com/paper/2307.11013/) (2023)
- [Deep-OSG: Deep Learning of Operators in Semigroup](https://mlsanity.com/paper/2302.03358/) (2023)
- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)
- [DUE: A Deep Learning Framework and Library for Modeling Unknown Equations](https://mlsanity.com/paper/2504.10373/) (2025)
- [Modeling Unknown Stochastic Dynamical System Subject to External Excitation](https://mlsanity.com/paper/2406.15747/) (2024)
- [Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learning](https://mlsanity.com/paper/2203.03458/) (2022)
- [Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systems](https://mlsanity.com/paper/2408.14821/) (2024)
- [Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamics](https://mlsanity.com/paper/2510.07549/) (2025)
- [Modeling unknown dynamical systems with hidden parameters](https://mlsanity.com/paper/2202.01858/) (2022)
- [Data-driven synchronization-avoiding algorithms in the explicit distributed structural analysis of soft tissue](https://mlsanity.com/paper/2207.02194/) (2022)
- [Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction](https://mlsanity.com/paper/2202.11432/) (2022)
- [Learning Fine Scale Dynamics from Coarse Observations via Inner Recurrence](https://mlsanity.com/paper/2206.01807/) (2022)
- [Physics-informed Multiple-Input Operators for efficient dynamic response prediction of structures](https://mlsanity.com/paper/2505.07090/) (2025)
- [Deep neural network based adaptive learning for switched systems](https://mlsanity.com/paper/2207.04623/) (2022)
- [Critical Sampling for Robust Evolution Operator Learning of Unknown Dynamical Systems](https://mlsanity.com/paper/2304.07485/) (2023)
- [Modeling Unknown Nonlocal PDE Systems via Flow Map Learning](https://mlsanity.com/paper/2608.00400/) (2026)
- [Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs](https://mlsanity.com/paper/2608.18148/) (2026)
- [Memory-Conditioned Diffusion Model for Generalized Langevin Dynamics](https://mlsanity.com/paper/2609.28371/) (2026)

## Abstract (arXiv)

We present a general numerical approach for constructing governing equations for unknown dynamical systems when only data on a subset of the state variables are available. The unknown equations for these observed variables are thus a reduced system of the complete set of state variables. Reduced systems possess memory integrals, based on the well known Mori-Zwanzig (MZ) formulism. Our numerical strategy to recover the reduced system starts by formulating a discrete approximation of the memory integral in the MZ formulation. The resulting unknown approximate MZ equations are of finite dimensional, in the sense that a finite number of past history data are involved. We then present a deep neural network structure that directly incorporates the history terms to produce memory in the network. The approach is suitable for any practical systems with finite memory length. We then use a set of numerical examples to demonstrate the effectiveness of our method.