# Model reduction for the material point method via an implicit neural representation of the deformation map

arXiv: https://arxiv.org/abs/2109.12390 · HTML: https://mlsanity.com/paper/2109.12390/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 144 references that are in the map.
- First posted on arXiv 25 Sep 2021.

## Builds on (most cited first, 25 of 25)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [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)
- [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)
- [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)
- [DiffTaichi: Differentiable Programming for Physical Simulation](https://mlsanity.com/paper/1910.00935/) (2019)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [ChainQueen: A Real-Time Differentiable Physical Simulator for Soft Robotics](https://mlsanity.com/paper/1810.01054/) (2018)
- [Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow](https://mlsanity.com/paper/1802.10123/) (2018)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [High-order Differentiable Autoencoder for Nonlinear Model Reduction](https://mlsanity.com/paper/2102.11026/) (2021)
- [Learning to Generate Chairs, Tables and Cars with Convolutional Networks](https://mlsanity.com/paper/1411.5928/) (2014)
- [Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics](https://mlsanity.com/paper/1401.8044/) (2014)
- [Realtime Simulation of Thin-Shell Deformable Materials using CNN-Based Mesh Embedding](https://mlsanity.com/paper/1909.12354/) (2019)
- [Soft Hybrid Aerial Vehicle via Bistable Mechanism](https://mlsanity.com/paper/2011.00426/) (2020)
- [Hierarchical Optimization Time Integration for CFL-rate MPM Stepping](https://mlsanity.com/paper/1911.07913/) (2019)
- [A hybrid material-point spheropolygon-element method for solid and granular material interaction](https://mlsanity.com/paper/1909.13655/) (2019)

## Built on by (most cited first, 11 of 11)

- [CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations](https://mlsanity.com/paper/2206.02607/) (2022)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [Neural Stress Fields for Reduced-order Elastoplasticity and Fracture](https://mlsanity.com/paper/2310.17790/) (2023)
- [Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks](https://mlsanity.com/paper/2310.04867/) (2023)
- [Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equations](https://mlsanity.com/paper/2404.01145/) (2024)
- [Accelerate Neural Subspace-Based Reduced-Order Solver of Deformable Simulation by Lipschitz Optimization](https://mlsanity.com/paper/2409.03807/) (2024)
- [NeuROK: Generative 4D Neural Object Kinematics](https://mlsanity.com/paper/2605.30347/) (2026)
- [Near-realtime Facial Animation by Deep 3D Simulation Super-Resolution](https://mlsanity.com/paper/2305.03216/) (2023)
- [Neural Kinematic Bases for Fluids](https://mlsanity.com/paper/2504.15657/) (2025)
- [Reduced-order Neural Modeling with Differentiable Simulation for High-Detail Tactile Perception](https://mlsanity.com/paper/2605.05053/) (2026)
- [OnlyDense: Reduced-Order Modeling for Lagrangian simulation](https://mlsanity.com/paper/2606.09065/) (2026)

## Abstract (arXiv)

This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates the textitkinematics by approximating the deformation map using an implicit neural representation that restricts deformation trajectories to reside on a low-dimensional manifold. By explicitly approximating the deformation map, its spatiotemporal gradients -- in particular the deformation gradient and the velocity -- can be computed via analytical differentiation. In contrast to typical model-reduction techniques that construct a linear or nonlinear manifold to approximate the (finite number of) degrees of freedom characterizing a given spatial discretization, the use of an implicit neural representation enables the proposed method to approximate the textitcontinuous deformation map. This allows the kinematic approximation to remain agnostic to the discretization. Consequently, the technique supports dynamic discretizations -- including resolution changes -- during the course of the online reduced-order-model simulation. To generate textitdynamics for the generalized coordinates, we propose a family of projection techniques. At each time step, these techniques: (1) Calculate full-space kinematics at quadrature points, (2) Calculate the full-space dynamics for a subset of `sample' material points, and (3) Calculate the reduced-space dynamics by projecting the updated full-space position and velocity onto the low-dimensional manifold and tangent space, respectively. We achieve significant computational speedup via hyper-reduction that ensures all three steps execute on only a small subset of the problem's spatial domain. Large-scale numerical examples with millions of material points illustrate the method's ability to gain an order of magnitude computational-cost saving -- indeed textitreal-time simulations -- with negligible errors.