# Data-Driven Deep Learning of Partial Differential Equations in Modal Space

arXiv: https://arxiv.org/abs/1910.06948 · HTML: https://mlsanity.com/paper/1910.06948/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Oct 2019.

## Built on by (most cited first, 30 of 46)

- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains](https://mlsanity.com/paper/2104.10873/) (2021)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [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)
- [tLaSDI: Thermodynamics-informed latent space dynamics identification](https://mlsanity.com/paper/2403.05848/) (2024)
- [Robust discovery of partial differential equations in complex situations](https://mlsanity.com/paper/2106.00008/) (2021)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data](https://mlsanity.com/paper/2205.13545/) (2022)
- [WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming](https://mlsanity.com/paper/2211.03134/) (2022)
- [Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning](https://mlsanity.com/paper/2405.09285/) (2024)
- [The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation](https://mlsanity.com/paper/2103.11488/) (2021)
- [A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks](https://mlsanity.com/paper/2210.04338/) (2022)
- [Energy-Dissipative Evolutionary Deep Operator Neural Networks](https://mlsanity.com/paper/2306.06281/) (2023)
- [Online Weak-form Sparse Identification of Partial Differential Equations](https://mlsanity.com/paper/2203.03979/) (2022)
- [Fourier neural operator for real-time simulation of 3D dynamic urban microclimate](https://mlsanity.com/paper/2308.03985/) (2023)
- [Deep-OSG: Deep Learning of Operators in Semigroup](https://mlsanity.com/paper/2302.03358/) (2023)
- [DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method](https://mlsanity.com/paper/2111.01356/) (2021)
- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)
- [Finite Expression Methods for Discovering Physical Laws from Data](https://mlsanity.com/paper/2305.08342/) (2023)
- [DUE: A Deep Learning Framework and Library for Modeling Unknown Equations](https://mlsanity.com/paper/2504.10373/) (2025)
- [FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning](https://mlsanity.com/paper/2509.12344/) (2025)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [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)
- [Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?](https://mlsanity.com/paper/2412.06158/) (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)
- [Adaptive deep density approximation for stochastic dynamical systems](https://mlsanity.com/paper/2405.02810/) (2024)

## Abstract (arXiv)

We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of the underlying PDE numerically. The evolution operator of the PDE, defined in infinite-dimensional space, maps the solution from a current time to a future time and completely characterizes the solution evolution of the underlying unknown PDE. Our recovery strategy relies on approximation of the evolution operator in a properly defined modal space, i.e., generalized Fourier space, in order to reduce the problem to finite dimensions. The finite dimensional approximation is then accomplished by training a deep neural network structure, which is based on residual network (ResNet), using the given data. Error analysis is provided to illustrate the predictive accuracy of the proposed method. A set of examples of different types of PDEs, including inviscid Burgers' equation that develops discontinuity in its solution, are presented to demonstrate the effectiveness of the proposed method.