# Conditionally Parameterized, Discretization-Aware Neural Networks for Mesh-Based Modeling of Physical Systems

arXiv: https://arxiv.org/abs/2109.09510 · HTML: https://mlsanity.com/paper/2109.09510/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 44 references that are in the map.
- First posted on arXiv 15 Sep 2021.

## Builds on (most cited first, 18 of 18)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Spatial Temporal Graph Convolutional Networks for Skeleton-Based Action Recognition](https://mlsanity.com/paper/1801.07455/) (2018)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Large-scale Point Cloud Semantic Segmentation with Superpoint Graphs](https://mlsanity.com/paper/1711.09869/) (2017)
- [Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs](https://mlsanity.com/paper/1704.02901/) (2017)
- [CondConv: Conditionally Parameterized Convolutions for Efficient Inference](https://mlsanity.com/paper/1904.04971/) (2019)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids](https://mlsanity.com/paper/1810.01566/) (2018)
- [GaAN: Gated Attention Networks for Learning on Large and Spatiotemporal Graphs](https://mlsanity.com/paper/1803.07294/) (2018)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics](https://mlsanity.com/paper/1912.11114/) (2019)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Machine Learning-augmented Predictive Modeling of Turbulent Separated Flows over Airfoils](https://mlsanity.com/paper/1608.03990/) (2016)

## Built on by (most cited first, 17 of 17)

- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [A Finite Element-Inspired Hypergraph Neural Network: Application to Fluid Dynamics Simulations](https://mlsanity.com/paper/2212.14545/) (2022)
- [Mesh-based Super-Resolution of Fluid Flows with Multiscale Graph Neural Networks](https://mlsanity.com/paper/2409.07769/) (2024)
- [Predicting fluid-structure interaction with graph neural networks](https://mlsanity.com/paper/2210.04193/) (2022)
- [A composable machine-learning approach for steady-state simulations on high-resolution grids](https://mlsanity.com/paper/2210.05837/) (2022)
- [Interpretable A-posteriori Error Indication for Graph Neural Network Surrogate Models](https://mlsanity.com/paper/2311.07548/) (2023)
- [Multi-scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators](https://mlsanity.com/paper/2206.14687/) (2022)
- [An extensible Benchmarking Graph-Mesh dataset for studying Steady-State Incompressible Navier-Stokes Equations](https://mlsanity.com/paper/2206.14709/) (2022)
- [Scalable and Consistent Graph Neural Networks for Distributed Mesh-based Data-driven Modeling](https://mlsanity.com/paper/2410.01657/) (2024)
- [Learning Effective Dynamics across Spatio-Temporal Scales of Complex Flows](https://mlsanity.com/paper/2502.07990/) (2025)
- [FIGNN: Feature-Specific Interpretability for Graph Neural Network Surrogate Models](https://mlsanity.com/paper/2506.11398/) (2025)
- [A Mesh-Adaptive Hypergraph Neural Network for Unsteady Flow Around Oscillating and Rotating Structures](https://mlsanity.com/paper/2503.22252/) (2025)
- [Simulating Network Paths with Recurrent Buffering Units](https://mlsanity.com/paper/2202.13870/) (2022)
- [A domain decomposition-based autoregressive deep learning model for unsteady and nonlinear partial differential equations](https://mlsanity.com/paper/2408.14461/) (2024)
- [Physics-Refined Spatiotemporal Forecasting on Open-Boundary Hydrologic Graphs](https://mlsanity.com/paper/2610.01765/) (2026)
- [Beyond Time Induction: Structured Conditional Neural Solvers for Parametric Stochastic-Volatility Pricing](https://mlsanity.com/paper/2610.19254/) (2026)

## Abstract (arXiv)

Simulations of complex physical systems are typically realized by discretizing partial differential equations (PDEs) on unstructured meshes. While neural networks have recently been explored for surrogate and reduced order modeling of PDE solutions, they often ignore interactions or hierarchical relations between input features, and process them as concatenated mixtures. We generalize the idea of conditional parameterization -- using trainable functions of input parameters to generate the weights of a neural network, and extend them in a flexible way to encode critical information. Inspired by discretized numerical methods, choices of the parameters include physical quantities and mesh topology features. The functional relation between the modeled features and the parameters is built into the network architecture. The method is implemented on different networks and applied to frontier scientific machine learning tasks including the discovery of unmodeled physics, super-resolution of coarse fields, and the simulation of unsteady flows with chemical reactions. The results show that the conditionally-parameterized networks provide superior performance compared to their traditional counterparts. The CP-GNet - an architecture that can be trained on very few data snapshots - is proposed as the first deep learning model capable of standalone prediction of reacting flows on irregular meshes.