Dynamic Systems & Deep Learning

A collection of resources that combine dynamical systems, control with deep learning. Please feel free to submit a pull request if you want to add good papers. Welcome to contact me for cooperation if there are many overlaps in our research interests。

Differential Equations in Deep Learning

General Architectures

• Recurrent Neural Networks for Multivariate Time Series with Missing Values: Scientific Reports18

• Deep Equilibrium Models: NeurIPS19

• Fast and Deep Graph Neural Networks: AAAI20

Neural ODEs

• Neural Ordinary Differential Equations: NeurIPS18
• Augmented Neural ODEs: NeurIPS19
• Dissecting Neural ODEs: arXiv20
• Latent ODEs for Irregularly-Sampled Time Series: arXiv19
• Learning unknown ODE models with Gaussian processes: arXiv18
• ODE2VAE: Deep generative second order ODEs with Bayesian neural networks: NeurIPS19
• Stable Neural Flows: arXiv20
• On Second Order Behaviour in Augmented Neural ODEs arXiv20
• Snode: Spectral discretization of neural odes for system identification arXiv19
• Learning Differential Equations that are Easy to Solve NeurIPS20 code
• An Ode to an ODE arXiv20
• ANODEV2: A Coupled Neural ODE Evolution Framework arXiv19
• Neural Manifold Ordinary Differential Equations. NeurIPS20
• Neural Delay Differential Equations ICLR21
• Neural Jump Ordinary Differential Equations ICLR21
• Learning Neural Event Functions for Ordinary Differential Equations. ICLR21
• Neural ODE Processes. ICLR21
• ResNet After All: Neural ODEs and Their Numerical Solution. ICLR21
• Go with the flow: Adaptive control for Neural ODEs. ICLR21
• MALI: A memory efficient and reverse accurate integrator for Neural ODEs. ICLR21
• On the Verification of Neural ODEs with Stochastic Guarantees. AAAI21
• Forecasting Reservoir Inflow via Recurrent Neural ODEs. [AAAI21]

Speed up Training of Neural ODEs

• Accelerating Neural ODEs with Spectral Elements: arXiv19
• Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE: ICML20
• "Hey, that's not an ODE": Faster ODE Adjoints with 12 Lines of Code arXiv20
• How to Train you Neural ODE: arXiv20
• Hypersolvers: Toward Fast Continuous-Depth Models NeurIPS20
• Interpolation technique to speed up gradients propagation in Neural Ordinary Differential Equations. NeurIPS20

Neural SDEs

• Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise: arXiv19
• Neural Jump Stochastic Differential Equations: arXiv19
• Towards Robust and Stable Deep Learning Algorithms for Forward Backward Stochastic Differential Equations: arXiv19
• Scalable Gradients and Variational Inference for Stochastic Differential Equations AISTATS20
• Score-Based Generative Modeling through Stochastic Differential Equations. ICLR21
• Neural SDEs Made Easy: SDEs are Infinite-Dimensional GANs url

Neural CSDEs

• Learning Continuous-Time Dynamics by Stochastic Differential Networks. url

Neural CDEs

• Neural Controlled Differential Equations for Irregular Time Series: ArXiv2020
• Neural CDEs for Long Time Series via the Log-ODE Method: ArXiv2020

Normalizing Flows

• Monge-Ampère Flow for Generative Modeling: arXiv18
• FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models: ICLR19
• Equivariant Flows: sampling configurations for multi-body systems with symmetric energies: arXiv18
• OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport. [AAAI21]
• Accelerating Continuous Normalizing Flow with Trajectory Polynomial Regularization. [AAAI21]

Applications

• Learning Dynamics of Attention: Human Prior for Interpretable Machine Reasoning: NeurIPS19
• Graph Neural Ordinary Differential Equations arXiv19
• Continuous graph neural networks ICML2020
• Neural Dynamics on Complex Networks arXiv19

Energy based models

Hamilton

• Hamiltonian Neural Networks: NeurIPS19 code
• Hamiltonian generative networksICLR2020 code
• Sparse Symplectically Integrated Neural Networks NeurIPS20 code
• Nonseparable symplectic neural networks code
• SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems arxiV20
• Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control: arXiv19
• Symplectic Recurrent Neural Network arXiv19
• Conformation-Guided Molecular Representation with Hamiltonian Neural Networks. ICLR21
• Identifying Physical Law of Hamiltonian Systems via Meta-Learning. ICLR21

Applications

• Hamiltonian graph networks with ode integrators arXiv19

Lagrange

• Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning: ICLR19
• Unsupervised Learning of Lagrangian Dynamics from Images for Prediction and Control NeurIPS20
• Lagrangian Neural Networks: ICLR20 DeepDiffEq

Deep Learning Methods for Differential Equations

Solving Differential Equations

Learning PDEs

• PDE-Net: Learning PDEs From Data: ICML18
• PDE-Net 2.0: Learning PDEs from Data Journal of Computational Physics
• Solving parametric PDE problems with artificial neural networks arXiv
• Multipole Graph Neural Operator for Parametric Partial Differential Equations. NeurIPS20
• Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning NeurIPS20
• A neural method for symbolically solving partial differential equations url
• Neural Partial Differential Equations with Functional Convolution. url
• Fourier Neural Operator for Parametric Partial Differential Equations. ICLR21
• Learning continuous-time PDEs from sparse data with graph neural networks. ICLR21

Applications

• PDE-regularized Neural Networks for Image Classification.url
• PDE-Driven Spatiotemporal Disentanglement. ICLR21

Model Discovery

• Universal Differential Equations for Scientific Machine Learning: arXiv20

Deep Control

Model-Predictive-Control

• Differentiable MPC for End-to-end Planning and Control: NeurIPS18
• Neural lyapunov model predictive control arXiv20

Dynamical System View of Deep Learning

Recurrent Neural Networks

• A Comprehensive Review of Stability Analysis of Continuous-Time Recurrent Neural Networks: IEEE Transactions on Neural Networks 2006
• AntysimmetricRNN: A Dynamical System View on Recurrent Neural Networks: ICLR19
• Recurrent Neural Networks in the Eye of Differential Equations: arXiv19
• Visualizing memorization in RNNs: distill19
• One step back, two steps forward: interference and learning in recurrent neural networks: arXiv18
• Reverse engineering recurrent networks for sentiment classification reveals line attractor dynamics: arXiv19
• System Identification with Time-Aware Neural Sequence Models: AAAI20
• Universality and Individuality in recurrent networks: NeurIPS19
• The LOB Recreation Model: Predicting the Limit Order Book from TAQ History Using an Ordinary Differential Equation Recurrent Neural Network. [AAAI21]
• Lipschitz Recurrent Neural Networks. ICLR21

Theory and Perspectives

• Deep information propagation arXiv16
• A mean field optimal control formulation of deep learning Research in Mathematical Science
• A Proposal on Machine Learning via Dynamical Systems: Communications in Mathematics and Statistics 2017
• Deep learning as optimal control problems:models and numerical methods arXiv19
• Deep Learning Theory Review: An Optimal Control and Dynamical Systems Perspective: arXiv19
• Stable Architectures for Deep Neural Networks: IP17
• Beyond Finite Layer Neural Network: Bridging Deep Architects and Numerical Differential Equations: ICML18
• Review: Ordinary Differential Equations For Deep Learning: arXiv19
• Universal approximation power of deep residual neural networks via nonlinear control theory. ICLR21
• Physics-aware, probabilistic model order reduction with guaranteed stability. ICLR21
• Dynamics of Deep Equilibrium Linear Models. ICLR21
• Identifying nonlinear dynamical systems with multiple time scales and long-range dependencies . ICLR21

Optimization

• Gradient and Hamiltonian Dynamics Applied to Learning in Neural Networks: NIPS96
• Maximum Principle Based Algorithms for Deep Learning: JMLR17
• Hamiltonian Descent Methods: arXiv18
• Port-Hamiltonian Approach to Neural Network Training: CDC19, code
• An Optimal Control Approach to Deep Learning and Applications to Discrete-Weight Neural Networks: arXiv19
• Optimizing Millions of Hyperparameters by Implicit Differentiation: arXiv19
• Shadowing Properties of Optimization Algorithms: NeurIPS19
• NOVAS: Non-convex Optimization via Adaptive Stochastic Search for End-to-end Learning and Control. ICLR21
• A Hybrid Stochastic Gradient Hamiltonian Monte Carlo Method. [AAAI21]

Signals and systems

• Multiplicative Filter Networks. ICLR21

Others

• Training Generative Adversarial Networks by Solving Ordinary Differential Equations. NeurIPS20
• Variational Intrinsic Control Revisited. ICLR21
• ECG ODE-GAN: Learning Ordinary Differential Equations of ECG Dynamics via Generative Adversarial Learning [AAAI21]

Software and Libraries

Python

• torchdyn: PyTorch library for all things neural differential equations. repo, docs
• torchdiffeq: Differentiable ODE solvers with full GPU support and O(1)-memory backpropagation: repo
• torchsde: Stochastic differential equation (SDE) solvers with GPU support and efficient sensitivity analysis: repo
• torchSODE: PyTorch Block-Diagonal ODE solver: repo
• JAX MD: JAX MD: A Framework for Differentiable Physics. repo NeurIPS20

Julia

• DiffEqFlux: Neural differential equation solvers with O(1) backprop, GPUs, and stiff+non-stiff DE solvers. Supports stiff and non-stiff neural ordinary differential equations (neural ODEs), neural stochastic differential equations (neural SDEs), neural delay differential equations (neural DDEs), neural partial differential equations (neural PDEs), and neural jump stochastic differential equations (neural jump diffusions). All of these can be solved with high order methods with adaptive time-stepping and automatic stiffness detection to switch between methods. repo

• NeuralNetDiffEq: Implementations of ODE, SDE, and PDE solvers via deep neural networks: repo

Websites and Blogs

• Scientific ML Blog (Chris Rackauckas and SciML): link

Slides

• NeurIPS20 Implicit Layers

• ODE Talk