# Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs

arXiv: https://arxiv.org/abs/2601.11259 · HTML: https://mlsanity.com/paper/2601.11259/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 54 of its 93 references that are in the map.
- First posted on arXiv 16 Jan 2026.

## Builds on (most cited first, 30 of 54)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders](https://mlsanity.com/paper/1802.03480/) (2018)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [Learning continuous-time PDEs from sparse data with graph neural networks](https://mlsanity.com/paper/2006.08956/) (2020)
- [PyG 2.0: Scalable Learning on Real World Graphs](https://mlsanity.com/paper/2507.16991/) (2025)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Equivariant Graph Neural Operator for Modeling 3D Dynamics](https://mlsanity.com/paper/2401.11037/) (2024)
- [Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers](https://mlsanity.com/paper/2008.09768/) (2020)
- [Thermodynamics-informed graph neural networks](https://mlsanity.com/paper/2203.01874/) (2022)
- [Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions](https://mlsanity.com/paper/2211.06786/) (2022)
- [Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall](https://mlsanity.com/paper/2212.05023/) (2022)
- [A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations](https://mlsanity.com/paper/2103.06183/) (2021)
- [SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases](https://mlsanity.com/paper/2308.13212/) (2023)
- [Regularized linear autoencoders recover the principal components, eventually](https://mlsanity.com/paper/2007.06731/) (2020)
- [Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks](https://mlsanity.com/paper/2504.02843/) (2025)
- [Reduced Order Modeling with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2502.10930/) (2025)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)

## Abstract (arXiv)

Graph Neural Networks (GNNs) are emerging as powerful tools for nonlinear Model Order Reduction (MOR) of time-dependent parameterized Partial Differential Equations (PDEs). However, existing methodologies struggle to combine geometric inductive biases with interpretable latent behavior, overlooking dynamics-driven features or disregarding spatial information. In this work, we address this gap by introducing Latent Dynamics Graph Convolutional Network (LD-GCN), a purely data-driven, encoder-free architecture that learns a global, low-dimensional representation of dynamical systems conditioned on external inputs and parameters. The temporal evolution is modeled in the latent space and advanced through time-stepping, allowing for time-extrapolation, and the trajectories are consistently decoded onto geometrically parameterized domains using a GNN. Our framework enhances interpretability by enabling the analysis of the reduced dynamics and supporting zero-shot prediction through latent interpolation. The methodology is mathematically validated via a universal approximation theorem for encoder-free architectures, and numerically tested on complex computational mechanics problems involving physical and geometric parameters, including the detection of bifurcating phenomena for Navier-Stokes equations. Code availability: https://github.com/lorenzotomada/ld-gcn-rom