# High-order Differentiable Autoencoder for Nonlinear Model Reduction

arXiv: https://arxiv.org/abs/2102.11026 · HTML: https://mlsanity.com/paper/2102.11026/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 81 references that are in the map.
- First posted on arXiv 19 Feb 2021.

## Builds on (most cited first, 10 of 10)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [End-to-end Optimized Image Compression](https://mlsanity.com/paper/1611.01704/) (2016)
- [Adversarial Autoencoders](https://mlsanity.com/paper/1511.05644/) (2015)
- [Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow](https://mlsanity.com/paper/1802.10123/) (2018)
- [Video Compression With Rate-Distortion Autoencoders](https://mlsanity.com/paper/1908.05717/) (2019)
- [Complementary Dynamics](https://mlsanity.com/paper/2009.02462/) (2020)

## Built on by (most cited first, 30 of 32)

- [TextDeformer: Geometry Manipulation using Text Guidance](https://mlsanity.com/paper/2304.13348/) (2023)
- [Human Trajectory Prediction via Neural Social Physics](https://mlsanity.com/paper/2207.10435/) (2022)
- [CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations](https://mlsanity.com/paper/2206.02607/) (2022)
- [Neural Cloth Simulation](https://mlsanity.com/paper/2212.11220/) (2022)
- [Neural Jacobian Fields: Learning Intrinsic Mappings of Arbitrary Meshes](https://mlsanity.com/paper/2205.02904/) (2022)
- [Neural Stress Fields for Reduced-order Elastoplasticity and Fracture](https://mlsanity.com/paper/2310.17790/) (2023)
- [ContourCraft: Learning to Resolve Intersections in Neural Multi-Garment Simulations](https://mlsanity.com/paper/2405.09522/) (2024)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Dress-1-to-3: Single Image to Simulation-Ready 3D Outfit with Diffusion Prior and Differentiable Physics](https://mlsanity.com/paper/2502.03449/) (2025)
- [Data-Free Learning of Reduced-Order Kinematics](https://mlsanity.com/paper/2305.03846/) (2023)
- [Model reduction for the material point method via an implicit neural representation of the deformation map](https://mlsanity.com/paper/2109.12390/) (2021)
- [RTG-SLAM: Real-time 3D Reconstruction at Scale using Gaussian Splatting](https://mlsanity.com/paper/2404.19706/) (2024)
- [Fine-grained differentiable physics: a yarn-level model for fabrics](https://mlsanity.com/paper/2202.00504/) (2022)
- [Differentiable Hybrid Traffic Simulation](https://mlsanity.com/paper/2210.08046/) (2022)
- [Human Trajectory Forecasting with Explainable Behavioral Uncertainty](https://mlsanity.com/paper/2307.01817/) (2023)
- [Neural Autoencoder-Based Structure-Preserving Model Order Reduction and Control Design for High-Dimensional Physical Systems](https://mlsanity.com/paper/2312.06256/) (2023)
- [Low-dimensional Data-based Surrogate Model of a Continuum-mechanical Musculoskeletal System Based on Non-intrusive Model Order Reduction](https://mlsanity.com/paper/2302.06528/) (2023)
- [Neural Modes: Self-supervised Learning of Nonlinear Modal Subspaces](https://mlsanity.com/paper/2404.17620/) (2024)
- [PoissonNet: A Local-Global Approach for Learning on Surfaces](https://mlsanity.com/paper/2510.14146/) (2025)
- [Human Motion Prediction under Unexpected Perturbation](https://mlsanity.com/paper/2403.15891/) (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)
- [Bayesian Differentiable Physics for Cloth Digitalization](https://mlsanity.com/paper/2402.17664/) (2024)
- [Learning Extremely High Density Crowds as Active Matters](https://mlsanity.com/paper/2503.12168/) (2025)
- [Near-realtime Facial Animation by Deep 3D Simulation Super-Resolution](https://mlsanity.com/paper/2305.03216/) (2023)
- [Spatially Adaptive Cloth Regression with Implicit Neural Representations](https://mlsanity.com/paper/2311.16344/) (2023)
- [Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback design](https://mlsanity.com/paper/2403.18044/) (2024)
- [PhysSkin: Real-Time and Generalizable Physics-Based Animation via Self-Supervised Neural Skinning](https://mlsanity.com/paper/2603.23194/) (2026)
- [Momentum-Conserving Graph Neural Networks for Deformable Objects](https://mlsanity.com/paper/2604.26097/) (2026)
- [Reduced-order Neural Modeling with Differentiable Simulation for High-Detail Tactile Perception](https://mlsanity.com/paper/2605.05053/) (2026)

## Abstract (arXiv)

This paper provides a new avenue for exploiting deep neural networks to improve physics-based simulation. Specifically, we integrate the classic Lagrangian mechanics with a deep autoencoder to accelerate elastic simulation of deformable solids. Due to the inertia effect, the dynamic equilibrium cannot be established without evaluating the second-order derivatives of the deep autoencoder network. This is beyond the capability of off-the-shelf automatic differentiation packages and algorithms, which mainly focus on the gradient evaluation. Solving the nonlinear force equilibrium is even more challenging if the standard Newton's method is to be used. This is because we need to compute a third-order derivative of the network to obtain the variational Hessian. We attack those difficulties by exploiting complex-step finite difference, coupled with reverse automatic differentiation. This strategy allows us to enjoy the convenience and accuracy of complex-step finite difference and in the meantime, to deploy complex-value perturbations as collectively as possible to save excessive network passes. With a GPU-based implementation, we are able to wield deep autoencoders (e.g., 10+ layers) with a relatively high-dimension latent space in real-time. Along this pipeline, we also design a sampling network and a weighting network to enable weight-varying Cubature integration in order to incorporate nonlinearity in the model reduction. We believe this work will inspire and benefit future research efforts in nonlinearly reduced physical simulation problems.