# Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics

arXiv: https://arxiv.org/abs/2205.03990 · HTML: https://mlsanity.com/paper/2205.03990/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 46 references that are in the map.
- First posted on arXiv 9 May 2022.

## Builds on (most cited first, 26 of 26)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures](https://mlsanity.com/paper/2002.10253/) (2020)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Deep Neural Networks for Data-Driven Turbulence Models](https://mlsanity.com/paper/1806.04482/) (2018)
- [DiscretizationNet: A Machine-Learning based solver for Navier-Stokes Equations using Finite Volume Discretization](https://mlsanity.com/paper/2005.08357/) (2020)
- [Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3D](https://mlsanity.com/paper/2012.11893/) (2020)
- [Hybrid FEM-NN models: Combining artificial neural networks with the finite element method](https://mlsanity.com/paper/2101.00962/) (2021)
- [Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data](https://mlsanity.com/paper/2001.04816/) (2020)
- [FEA-Net: A Physics-guided Data-driven Model for Efficient Mechanical Response Prediction](https://mlsanity.com/paper/2002.01893/) (2020)
- [Evolutionary Multi Objective Optimization Algorithm for Community Detection in Complex Social Networks](https://mlsanity.com/paper/2005.03181/) (2020)

## Built on by (most cited first, 30 of 44)

- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation](https://mlsanity.com/paper/2311.07896/) (2023)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction](https://mlsanity.com/paper/2504.19496/) (2025)
- [NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Data](https://mlsanity.com/paper/2305.18694/) (2023)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [P2C2Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics](https://mlsanity.com/paper/2411.00040/) (2024)
- [Multi-fidelity wavelet neural operator with application to uncertainty quantification](https://mlsanity.com/paper/2208.05606/) (2022)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [DiffHybrid-UQ: Uncertainty Quantification for Differentiable Hybrid Neural Modeling](https://mlsanity.com/paper/2401.00161/) (2023)
- [OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference](https://mlsanity.com/paper/2602.01493/) (2026)
- [Surrogate Neural Networks to Estimate Parametric Sensitivity of Ocean Models](https://mlsanity.com/paper/2311.08421/) (2023)
- [PAPM: A Physics-aware Proxy Model for Process Systems](https://mlsanity.com/paper/2407.05232/) (2024)
- [Probabilistic Physics-integrated Neural Differentiable Modeling for Isothermal Chemical Vapor Infiltration Process](https://mlsanity.com/paper/2311.07798/) (2023)
- [Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space](https://mlsanity.com/paper/2405.01975/) (2024)
- [A spatiotemporal deep learning framework for prediction of crack dynamics in heterogeneous solids: efficient mapping of concrete microstructures to its fracture properties](https://mlsanity.com/paper/2407.15665/) (2024)
- [Model-Agnostic AI Framework with Explicit Time Integration for Long-Term Fluid Dynamics Prediction](https://mlsanity.com/paper/2412.05657/) (2024)
- [MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation](https://mlsanity.com/paper/2501.15987/) (2025)
- [When do World Models Successfully Learn Dynamical Systems?](https://mlsanity.com/paper/2507.04898/) (2025)
- [Self-composing neural operators for high-frequency and multiscale PDE surrogates](https://mlsanity.com/paper/2508.20650/) (2025)
- [A Staged Deep Learning Approach to Spatial Refinement in 3D Temporal Atmospheric Transport](https://mlsanity.com/paper/2412.10945/) (2024)
- [Multi-fidelity Reinforcement Learning Control for Complex Dynamical Systems](https://mlsanity.com/paper/2504.05588/) (2025)
- [Topology-aware Neural Flux Prediction Guided by Physics](https://mlsanity.com/paper/2506.05676/) (2025)
- [Localized FNO for Spatiotemporal Hemodynamic Upsampling in Aneurysm MRI](https://mlsanity.com/paper/2507.13789/) (2025)
- [Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling](https://mlsanity.com/paper/2510.00233/) (2025)
- [Memory-Conditioned Flow-Matching for Stable Autoregressive PDE Rollouts](https://mlsanity.com/paper/2602.06689/) (2026)
- [Physics-Informed Neural Networks for Biological 2D+t Reaction-Diffusion Systems](https://mlsanity.com/paper/2604.18548/) (2026)
- [Learning to simulate partially known spatio-temporal dynamics with trainable difference operators](https://mlsanity.com/paper/2307.14395/) (2023)
- [Implicit Neural Differential Model for Spatiotemporal Dynamics](https://mlsanity.com/paper/2504.02260/) (2025)

## Abstract (arXiv)

Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by ``baking'' the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers', and Navier-Stokes equations.