# CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations

arXiv: https://arxiv.org/abs/2206.02607 · HTML: https://mlsanity.com/paper/2206.02607/

## Key facts

- Cited by 58 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 96 references that are in the map.
- First posted on arXiv 6 Jun 2022.

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Neural Sparse Voxel Fields](https://mlsanity.com/paper/2007.11571/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)
- [MetaSDF: Meta-learning Signed Distance Functions](https://mlsanity.com/paper/2006.09662/) (2020)
- [AutoInt: Automatic Integration for Fast Neural Volume Rendering](https://mlsanity.com/paper/2012.01714/) (2020)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Learning Smooth Neural Functions via Lipschitz Regularization](https://mlsanity.com/paper/2202.08345/) (2022)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [SAPE: Spatially-Adaptive Progressive Encoding for Neural Optimization](https://mlsanity.com/paper/2104.09125/) (2021)
- [Learning Signal-Agnostic Manifolds of Neural Fields](https://mlsanity.com/paper/2111.06387/) (2021)
- [NTopo: Mesh-free Topology Optimization using Implicit Neural Representations](https://mlsanity.com/paper/2102.10782/) (2021)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [High-order Differentiable Autoencoder for Nonlinear Model Reduction](https://mlsanity.com/paper/2102.11026/) (2021)
- [Experience report of physics-informed neural networks in fluid simulations: pitfalls and frustration](https://mlsanity.com/paper/2205.14249/) (2022)
- [Data-driven Discovery of Closure Models](https://mlsanity.com/paper/1803.09318/) (2018)
- [Model reduction for the material point method via an implicit neural representation of the deformation map](https://mlsanity.com/paper/2109.12390/) (2021)
- [Mesh sampling and weighting for the hyperreduction of nonlinear Petrov-Galerkin reduced-order models with local reduced-order bases](https://mlsanity.com/paper/2008.02891/) (2020)

## Built on by (most cited first, 30 of 58)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Operator Learning with Neural Fields: Tackling PDEs on General Geometries](https://mlsanity.com/paper/2306.07266/) (2023)
- [Continuous PDE Dynamics Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2209.14855/) (2022)
- [AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields](https://mlsanity.com/paper/2406.02176/) (2024)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Generative Latent Neural PDE Solver using Flow Matching](https://mlsanity.com/paper/2503.22600/) (2025)
- [Make-It-Animatable: An Efficient Framework for Authoring Animation-Ready 3D Characters](https://mlsanity.com/paper/2411.18197/) (2024)
- [Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulation](https://mlsanity.com/paper/2507.02608/) (2025)
- [Geometric Operator Learning with Optimal Transport](https://mlsanity.com/paper/2507.20065/) (2025)
- [CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equations](https://mlsanity.com/paper/2402.14646/) (2024)
- [Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations](https://mlsanity.com/paper/2406.03919/) (2024)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes](https://mlsanity.com/paper/2306.15630/) (2023)
- [A Hierarchical Spatial Transformer for Massive Point Samples in Continuous Space](https://mlsanity.com/paper/2311.04434/) (2023)
- [Thermodynamics-informed neural networks for physically realistic mixed reality](https://mlsanity.com/paper/2210.13414/) (2022)
- [Latent assimilation with implicit neural representations for unknown dynamics](https://mlsanity.com/paper/2309.09574/) (2023)
- [Accurate Differential Operators for Hybrid Neural Fields](https://mlsanity.com/paper/2312.05984/) (2023)
- [Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML](https://mlsanity.com/paper/2301.13331/) (2023)
- [Reduced-order modeling for parameterized PDEs via implicit neural representations](https://mlsanity.com/paper/2311.16410/) (2023)
- [FUSE: Fast Unified Simulation and Estimation for PDEs](https://mlsanity.com/paper/2405.14558/) (2024)
- [NeuralClothSim: Neural Deformation Fields Meet the Thin Shell Theory](https://mlsanity.com/paper/2308.12970/) (2023)
- [Foundation Models for Scientific Discovery: From Paradigm Enhancement to Paradigm Transition](https://mlsanity.com/paper/2510.15280/) (2025)
- [Advancing Generalization in PINNs through Latent-Space Representations](https://mlsanity.com/paper/2411.19125/) (2024)
- [Physics-informed reduced order model with conditional neural fields](https://mlsanity.com/paper/2412.05233/) (2024)
- [Learning Visual Feature-Based World Models via Residual Latent Action](https://mlsanity.com/paper/2605.07079/) (2026)
- [Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularization](https://mlsanity.com/paper/2402.15636/) (2024)
- [A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs](https://mlsanity.com/paper/2502.08683/) (2025)
- [ClothTransformer: Unified Latent-Space Transformers for Scalable Cloth Simulation](https://mlsanity.com/paper/2605.27852/) (2026)

## Abstract (arXiv)

The long runtime of high-fidelity partial differential equation (PDE) solvers makes them unsuitable for time-critical applications. We propose to accelerate PDE solvers using reduced-order modeling (ROM). Whereas prior ROM approaches reduce the dimensionality of discretized vector fields, our continuous reduced-order modeling (CROM) approach builds a low-dimensional embedding of the continuous vector fields themselves, not their discretization. We represent this reduced manifold using continuously differentiable neural fields, which may train on any and all available numerical solutions of the continuous system, even when they are obtained using diverse methods or discretizations. We validate our approach on an extensive range of PDEs with training data from voxel grids, meshes, and point clouds. Compared to prior discretization-dependent ROM methods, such as linear subspace proper orthogonal decomposition (POD) and nonlinear manifold neural-network-based autoencoders, CROM features higher accuracy, lower memory consumption, dynamically adaptive resolutions, and applicability to any discretization. For equal latent space dimension, CROM exhibits 79× and 49× better accuracy, and 39× and 132× smaller memory footprint, than POD and autoencoder methods, respectively. Experiments demonstrate 109× and 89× wall-clock speedups over unreduced models on CPUs and GPUs, respectively. Videos and codes are available on the project page: https://crom-pde.github.io