# A Score Filter Enhanced Data Assimilation Framework for Data-Driven Dynamical Systems

arXiv: https://arxiv.org/abs/2603.14863 · HTML: https://mlsanity.com/paper/2603.14863/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 83 references that are in the map.
- First posted on arXiv 16 Mar 2026.

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Designing Network Design Spaces](https://mlsanity.com/paper/2003.13678/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network](https://mlsanity.com/paper/1808.03314/) (2018)
- [LSTM: A Search Space Odyssey](https://mlsanity.com/paper/1503.04069/) (2015)
- [A Critical Review of Recurrent Neural Networks for Sequence Learning](https://mlsanity.com/paper/1506.00019/) (2015)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems](https://mlsanity.com/paper/2309.00983/) (2023)
- [Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2303.02243/) (2023)
- [One-shot learning for solution operators of partial differential equations](https://mlsanity.com/paper/2104.05512/) (2021)
- [Dynamical Tests of a Deep-Learning Weather Prediction Model](https://mlsanity.com/paper/2309.10867/) (2023)
- [Diffusion-Model-Assisted Supervised Learning of Generative Models for Density Estimation](https://mlsanity.com/paper/2310.14458/) (2023)
- [Dual Convolutional LSTM Network for Referring Image Segmentation](https://mlsanity.com/paper/2001.11561/) (2020)
- [Ensemble score filter with image inpainting for data assimilation in tracking surface quasi-geostrophic dynamics with partial observations](https://mlsanity.com/paper/2501.12419/) (2025)
- [A Mathematical Analysis of Neural Operator Behaviors](https://mlsanity.com/paper/2410.21481/) (2024)
- [A Scalable Real-Time Data Assimilation Framework for Predicting Turbulent Atmosphere Dynamics](https://mlsanity.com/paper/2407.12168/) (2024)
- [A Discriminative Approach to Bayesian Filtering with Applications to Human Neural Decoding](https://mlsanity.com/paper/1807.06173/) (2018)
- [A Kernel Learning Method for Backward SDE Filter](https://mlsanity.com/paper/2201.10600/) (2022)

## Built on by (most cited first, 3 of 3)

- [Generative Model Proposal based Particle Filtering for Data Assimilation](https://mlsanity.com/paper/2607.01012/) (2026)
- [FREESIA: Covariance-Aware Posterior Transport for Expressive and Scalable Data Assimilation](https://mlsanity.com/paper/2609.25085/) (2026)
- [AECSF: Adaptive Ensemble Conditional Score Filtering for High-Dimensional Nonlinear Data Assimilation](https://mlsanity.com/paper/2609.32411/) (2026)

## Abstract (arXiv)

We introduce a score-filter-enhanced data assimilation framework designed to reduce predictive uncertainty in machine learning (ML) models for data-driven dynamical system forecasting. Machine learning serves as an efficient numerical model for predicting dynamical systems. However, even with sufficient data, model uncertainty remains and accumulates over time, causing the long-term performance of ML models to deteriorate. To overcome this difficulty, we integrate data assimilation techniques into the training process to iteratively refine the model predictions by incorporating observational information. Specifically, we apply the Ensemble Score Filter (EnSF), a generative AI-based training-free diffusion model approach, for solving the data assimilation problem in high-dimensional nonlinear complex systems. This leads to a hybrid data assimilation-training framework that combines ML with EnSF to improve long-term predictive performance. We shall demonstrate that EnSF-enhanced ML can effectively reduce predictive uncertainty in ML-based Lorenz-96 system prediction and the Korteweg-De Vries (KdV) equation prediction.