# Learning Differential Operators for Interpretable Time Series Modeling

arXiv: https://arxiv.org/abs/2209.01491 · HTML: https://mlsanity.com/paper/2209.01491/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 60 references that are in the map.
- First posted on arXiv 3 Sep 2022.

## Builds on (most cited first, 30 of 39)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Enhancing the Locality and Breaking the Memory Bottleneck of Transformer on Time Series Forecasting](https://mlsanity.com/paper/1907.00235/) (2019)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [Connecting the Dots: Multivariate Time Series Forecasting with Graph Neural Networks](https://mlsanity.com/paper/2005.11650/) (2020)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Time Series Forecasting With Deep Learning: A Survey](https://mlsanity.com/paper/2004.13408/) (2020)
- [A Transformer-based Framework for Multivariate Time Series Representation Learning](https://mlsanity.com/paper/2010.02803/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Recurrent Neural Networks for Time Series Forecasting: Current Status and Future Directions](https://mlsanity.com/paper/1909.00590/) (2019)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Deep Transformer Models for Time Series Forecasting: The Influenza Prevalence Case](https://mlsanity.com/paper/2001.08317/) (2020)
- [Temporal Pattern Attention for Multivariate Time Series Forecasting](https://mlsanity.com/paper/1809.04206/) (2018)

## Built on by (most cited first, 7 of 7)

- [A Survey of Explainable Artificial Intelligence (XAI) in Financial Time Series Forecasting](https://mlsanity.com/paper/2407.15909/) (2024)
- [Correcting Auto-Differentiation in Neural-ODE Training](https://mlsanity.com/paper/2306.02192/) (2023)
- [Physics-Guided Learning of Meteorological Dynamics for Weather Downscaling and Forecasting](https://mlsanity.com/paper/2505.14555/) (2025)
- [Contrastive Prompt Clustering for Weakly Supervised Semantic Segmentation](https://mlsanity.com/paper/2508.17009/) (2025)
- [Interpretable time series neural representation for classification purposes](https://mlsanity.com/paper/2310.16696/) (2023)
- [Prime Convolutional Model: Breaking the Ground for Theoretical Explainability](https://mlsanity.com/paper/2503.02773/) (2025)
- [Modeling Dynamic Mixtures of Time-Delay Systems from Streaming Time Series](https://mlsanity.com/paper/2605.26191/) (2026)

## Abstract (arXiv)

Modeling sequential patterns from data is at the core of various time series forecasting tasks. Deep learning models have greatly outperformed many traditional models, but these black-box models generally lack explainability in prediction and decision making. To reveal the underlying trend with understandable mathematical expressions, scientists and economists tend to use partial differential equations (PDEs) to explain the highly nonlinear dynamics of sequential patterns. However, it usually requires domain expert knowledge and a series of simplified assumptions, which is not always practical and can deviate from the ever-changing world. Is it possible to learn the differential relations from data dynamically to explain the time-evolving dynamics? In this work, we propose an learning framework that can automatically obtain interpretable PDE models from sequential data. Particularly, this framework is comprised of learnable differential blocks, named P-blocks, which is proved to be able to approximate any time-evolving complex continuous functions in theory. Moreover, to capture the dynamics shift, this framework introduces a meta-learning controller to dynamically optimize the hyper-parameters of a hybrid PDE model. Extensive experiments on times series forecasting of financial, engineering, and health data show that our model can provide valuable interpretability and achieve comparable performance to state-of-the-art models. From empirical studies, we find that learning a few differential operators may capture the major trend of sequential dynamics without massive computational complexity.