mlsanityData to 7 Oct 2026

Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling

arXiv 2109.06407 ↗First posted 14 Sep 2021cs.LG cs.ROcited by 40

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Abstract (arXiv)

Effective inclusion of physics-based knowledge into deep neural network models of dynamical systems can greatly improve data efficiency and generalization. Such a-priori knowledge might arise from physical principles (e.g., conservation laws) or from the system's design (e.g., the Jacobian matrix of a robot), even if large portions of the system dynamics remain unknown. We develop a framework to learn dynamics models from trajectory data while incorporating a-priori system knowledge as inductive bias. More specifically, the proposed framework uses physics-based side information to inform the structure of the neural network itself, and to place constraints on the values of the outputs and the internal states of the model. It represents the system's vector field as a composition of known and unknown functions, the latter of which are parametrized by neural networks. The physics-informed constraints are enforced via the augmented Lagrangian method during the model's training. We experimentally demonstrate the benefits of the proposed approach on a variety of dynamical systems -- including a benchmark suite of robotics environments featuring large state spaces, non-linear dynamics, external forces, contact forces, and control inputs. By exploiting a-priori system knowledge during training, the proposed approach learns to predict the system dynamics two orders of magnitude more accurately than a baseline approach that does not include prior knowledge, given the same training dataset.

Key facts. Cited by 40 papers in the map (data through 7 Oct 2026). Builds on 38 of its 54 references that are in the map. First posted on arXiv 14 Sep 2021.

Builds on

38 of 54 references.

References · 38

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2019Hamiltonian Neural Networkscited by 702
  3. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  4. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  5. 2019Augmented Neural ODEscited by 396
  6. 2020Lagrangian Neural Networkscited by 388
  7. 2021Brax -- A Differentiable Physics Engine for Large Scale Rigid Body Simulationcited by 375
  8. 2016Benchmarking Deep Reinforcement Learning for Continuous Controlcited by 361
  9. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  10. 2017PDE-Net: Learning PDEs from Datacited by 304
  11. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  12. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
26 more
  1. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  2. 2019Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learningcited by 205
  3. 2017Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equationscited by 203
  4. 2019Symplectic ODE-Net: Learning Hamiltonian Dynamics with Controlcited by 183
  5. 2017Learning Koopman Invariant Subspaces for Dynamic Mode Decompositioncited by 156
  6. 2020Dissecting Neural ODEscited by 150
  7. 2019Hamiltonian Generative Networkscited by 136
  8. 2017Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systemscited by 125
  9. 2020Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraintscited by 102
  10. 2018Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systemscited by 88
  11. 2018Neural Lander: Stable Drone Landing Control using Learned Dynamicscited by 82
  12. 2020Learning Differential Equations that are Easy to Solvecited by 81
  13. 2020Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learningcited by 77
  14. 2020Lagrangian Duality for Constrained Deep Learningcited by 72
  15. 2020Structure-preserving neural networkscited by 54
  16. 2017Imposing Hard Constraints on Deep Networks: Promises and Limitationscited by 42
  17. 2020Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Datacited by 40
  18. 2020Unsupervised Learning of Lagrangian Dynamics from Images for Prediction and Controlcited by 38
  19. 2021Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Controlcited by 37
  20. 2021Extending Lagrangian and Hamiltonian Neural Networks with Differentiable Contact Modelscited by 33
  21. 2020Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanicscited by 32
  22. 2019Constrained Deep Networks: Lagrangian Optimization via Log-Barrier Extensionscited by 30
  23. 2020LagNetViP: A Lagrangian Neural Network for Video Predictioncited by 24
  24. 2020Structured Mechanical Models for Robot Learning and Controlcited by 21
  25. 2020Training neural networks under physical constraints using a stochastic augmented Lagrangian approachcited by 19
  26. 2014A Novel Augmented Lagrangian Approach for Inequalities and Convergent Any-Time Non-Central Updatescited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 6; 2023: 10; 2024: 9; 2025: 8; 2026: 6.

40 papers.

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  1. 2022Compositional Learning of Dynamical System Models Using Port-Hamiltonian Neural Networkscited by 17
  2. 2021Physically Consistent Neural Networks for building thermal modeling: theory and analysiscited by 16
  3. 2023Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methodscited by 14
  4. 2024On latent dynamics learning in nonlinear reduced order modelingcited by 14
  5. 2024Learning to Walk from Three Minutes of Real-World Data with Semi-structured Dynamics Modelscited by 14
  6. 2022Taylor-Lagrange Neural Ordinary Differential Equations: Toward Fast Training and Evaluation of Neural ODEscited by 13
  7. 2023Physics Constrained Motion Prediction with Uncertainty Quantificationcited by 11
  8. 2022Knowledge-augmented Deep Learning and Its Applications: A Surveycited by 10
  9. 2023How to Learn and Generalize From Three Minutes of Data: Physics-Constrained and Uncertainty-Aware Neural Stochastic Differential Equationscited by 10
  10. 2022Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)cited by 9
  11. 2024Generative AI and Process Systems Engineering: The Next Frontiercited by 8
  12. 2023Autonomous Drifting with 3 Minutes of Data via Learned Tire Modelscited by 7
28 more
  1. 2022Learning Robust State Observers using Neural ODEs (longer version)cited by 6
  2. 2022Guaranteed Conformance of Neurosymbolic Models to Natural Constraintscited by 6
  3. 2024Robust Biharmonic Skinning Using Geometric Fieldscited by 6
  4. 2023Learning Control-Oriented Dynamical Structure from Datacited by 5
  5. 2024Understanding the Difficulty of Solving Cauchy Problems with PINNscited by 5
  6. 2023Transfer learning-based physics-informed convolutional neural network for simulating flow in porous media with time-varying controlscited by 4
  7. 2024The Smart Buildings Control Suite: A Diverse Open Source Benchmark to Evaluate and Scale HVAC Control Policies for Sustainabilitycited by 4
  8. 2024Morphological-Symmetry-Equivariant Heterogeneous Graph Neural Network for Robotic Dynamics Learningcited by 4
  9. 2025Dynami-CAL GraphNet: A Physics-Informed Graph Neural Network Conserving Linear and Angular Momentum for Dynamical Systemscited by 3
  10. 2025Meta-Learning Online Dynamics Model Adaptation in Off-Road Autonomous Drivingcited by 3
  11. 2025Investigating Lagrangian Neural Networks for Infinite Horizon Planning in Quadrupedal Locomotioncited by 3
  12. 2023Actor-Critic Methods using Physics-Informed Neural Networks: Control of a 1D PDE Model for Fluid-Cooled Battery Packscited by 2
  13. 2023Physics-Informed Kernel Embeddings: Integrating Prior System Knowledge with Data-Driven Controlcited by 1
  14. 2024Neural Port-Hamiltonian Differential Algebraic Equations for Compositional Learning of Electrical Networkscited by 1
  15. 2025Orthogonal projection-based regularization for efficient model augmentationcited by 1
  16. 2025mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networkscited by 1
  17. 2025Newtonian and Lagrangian Neural Networks: A Comparison Towards Efficient Inverse Dynamics Identificationcited by 1
  18. 2025Artifacts of Numerical Integration in Learning Dynamical Systemscited by 1
  19. 2026AdamFLIP: Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN Trainingcited by 1
  20. 2023Operating data of a specific Aquatic Center as a Benchmark for dynamic model learning: search for a valid prediction model over an 8-hour horizoncited by 0
  21. 2023A Distributed ADMM-based Deep Learning Approach for Thermal Control in Multi-Zone Buildings under Demand Response Eventscited by 0
  22. 2024Multivariate Physics-Informed Convolutional Autoencoder for Anomaly Detection in Power Distribution Systems with High Penetration of DERscited by 0
  23. 2025Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structurescited by 0
  24. 2026Holographic generative flows with AdS/CFTcited by 0
  25. 2026Data-Driven Physics Embedded Dynamics with Predictive Control and Reinforcement Learning for Quadrupedscited by 0
  26. 2026Learning the Stellar Structure Equations via Self-supervised Physics-Informed Neural Networkscited by 0
  27. 2026Diagnosing Faults in Reinforcement Learning Simulators and World Models with Canonical Polynomial Invariantscited by 0
  28. 2026Grounding Time-Series Foundation Models in Digital Twin Topology for Predictive Maintenancecited by 0