Data-driven discovery of coordinates and governing equations

Abstract

The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam's razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom autoencoder to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system. We demonstrate this approach on several example high-dimensional dynamical systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. It is the first method of its kind to place the discovery of coordinates and models on an equal footing.

25 pages, 6 figures; added acknowledgments

References in corpus (4)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• Sparse Identification for Nonlinear Optical Communication Systems: SINO Method
• Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition

Cited by in corpus (110)

• Physics-informed learning of governing equations from scarce data
• Enhancing Computational Fluid Dynamics with Machine Learning
• Interpretable Scientific Discovery with Symbolic Regression: A Review
• Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics
• Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control
• SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics
• Unsupervised discovery of interpretable hyperelastic constitutive laws
• Convolutional neural network and long short-term memory based reduced order surrogate for minimal turbulent channel flow
• Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data
• Improving aircraft performance using machine learning: a review
• Autonomous Discovery of Unknown Reaction Pathways from Data by Chemical Reaction Neural Network
• A nonlocal physics-informed deep learning framework using the peridynamic differential operator
• Physics-guided deep learning framework for predictive modeling of the Reynolds stress anisotropy
• Sparse identification of nonlinear dynamics with low-dimensionalized flow representations
• Time-series learning of latent-space dynamics for reduced-order model closure
• Machine Learning on Neutron and X-Ray Scattering
• Assessment of unsteady flow predictions using hybrid deep learning based reduced order models
• Data-driven discovery of intrinsic dynamics
• Physics-constrained, low-dimensional models for MHD: First-principles and data-driven approaches
• Deep learning to discover and predict dynamics on an inertial manifold
• Learning Variational Data Assimilation Models and Solvers
• Multi-fidelity surrogate modeling using long short-term memory networks
• Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations
• LaSDI: Parametric Latent Space Dynamics Identification
• Learning hydrodynamic equations for active matter from particle simulations and experiments
• Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions
• Applications and Techniques for Fast Machine Learning in Science
• Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations
• On spike-and-slab priors for Bayesian equation discovery of nonlinear dynamical systems via sparse linear regression
• DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimization
• On the role of nonlinear correlations in reduced-order modeling
• Data-driven Nonlinear Model Reduction to Spectral Submanifolds in Mechanical Systems
• Sparse nonlinear models of chaotic electroconvection
• OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle
• Nonlinear proper orthogonal decomposition for convection-dominated flows
• From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction
• Bayesian operator inference for data-driven reduced-order modeling
• Deep Learning Enhanced Dynamic Mode Decomposition
• Deep Learning Models for Global Coordinate Transformations that Linearize PDEs
• Predicting the failure of two-dimensional silica glasses
• Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation
• Implicit Regularization and Momentum Algorithms in Nonlinearly Parameterized Adaptive Control and Prediction
• Learning Latent Dynamics for Partially-Observed Chaotic Systems
• gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification
• Discovering Sparse Interpretable Dynamics from Partial Observations
• Reducing the complexity of chemical networks via interpretable autoencoders
• Structured Time-Delay Models for Dynamical Systems with Connections to Frenet-Serret Frame
• Deep reconstruction of strange attractors from time series
• Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems
• Discovering transition phenomena from data of stochastic dynamical systems with Lévy noise
• Percolation on feature-enriched interconnected systems
• A Machine Learning Framework for Computing the Most Probable Paths of Stochastic Dynamical Systems
• Augmenting astrophysical scaling relations with machine learning: application to reducing the Sunyaev-Zeldovich flux-mass scatter
• Reconstructing data-driven governing equations for cell phenotypic transitions: integration of data science and systems biology
• Discovering Conservation Laws using Optimal Transport and Manifold Learning
• Machine-Learning Non-Conservative Dynamics for New-Physics Detection
• Towards out of distribution generalization for problems in mechanics
• Data-driven discovery and extrapolation of parameterized pattern-forming dynamics
• Data-driven Modeling of Rotating Detonation Waves
• Double Diffusion Maps and their Latent Harmonics for Scientific Computations in Latent Space
• Model selection of chaotic systems from data with hidden variables using sparse data assimilation
• Learning the temporal evolution of multivariate densities via normalizing flows
• Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population
• Generative ODE Modeling with Known Unknowns
• Extracting Forces from Noisy Dynamics in Dusty Plasmas
• Describing condensed matter from atomically resolved imaging data: from structure to generative and causal models
• Deeptime: a Python library for machine learning dynamical models from time series data
• A hybrid learning method for system identification and optimal control
• Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas
• Deep Learning of Conjugate Mappings
• Improved Surrogate Modeling of Fluid Dynamics with Physics-Informed Neural Networks
• Deep Kernel Learning of Dynamical Models from High-Dimensional Noisy Data
• Reconstructing three-dimensional bluff body wake from sectional flow fields with convolutional neural networks
• System Identification Through Lipschitz Regularized Deep Neural Networks
• Data-driven discovery of multiscale chemical reactions governed by the law of mass action
• Virtual twins of nonlinear vibrating multiphysics microstructures: physics-based versus deep learning-based approaches
• Extracting Non-Gaussian Governing Laws from Data on Mean Exit Time
• Identifying nonlinear dynamical systems with multiple time scales and long-range dependencies
• Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks
• Random Feature Models for Learning Interacting Dynamical Systems
• Data-driven reduced order models using invariant foliations, manifolds and autoencoders
• Sparse Symplectically Integrated Neural Networks
• Metalearning generalizable dynamics from trajectories
• Sparsely constrained neural networks for model discovery of PDEs
• Promoting global stability in data-driven models of quadratic nonlinear dynamics
• CINDy: Conditional gradient-based Identification of Non-linear Dynamics -- Noise-robust recovery
• Learning a Reduced Basis of Dynamical Systems using an Autoencoder
• KuraNet: Systems of Coupled Oscillators that Learn to Synchronize
• Neural Ordinary Differential Equations for Data-Driven Reduced Order Modeling of Environmental Hydrodynamics
• Variational Deep Learning for the Identification and Reconstruction of Chaotic and Stochastic Dynamical Systems from Noisy and Partial Observations
• Discovery of slow variables in a class of multiscale stochastic systems via neural networks
• Rapidly Encoding Generalizable Dynamics in a Euclidean Symmetric Neural Network
• Kernel-based parameter estimation of dynamical systems with unknown observation functions
• Calibrating multi-dimensional complex ODE from noisy data via deep neural networks
• Spiral-wave dynamics in excitable media: Insights from dynamic mode decomposition
• Physics-Guided Discovery of Highly Nonlinear Parametric Partial Differential Equations
• Discovering conservation laws from trajectories via machine learning
• Model order reduction with neural networks: Application to laminar and turbulent flows
• Deep learning reveals hidden interactions in complex systems
• Localization of Invariable Sparse Errors in Dynamic Systems
• Testing topological conjugacy of time series
• Reverse engineering learned optimizers reveals known and novel mechanisms
• A quantum inspired approach to learning dynamical laws from data -- block-sparsity and gauge-mediated weight sharing
• Stability of a Parametrically Driven, Coupled Oscillator System: An Auxillary Function Method Approach
• Learning Low-Dimensional Quadratic-Embeddings of High-Fidelity Nonlinear Dynamics using Deep Learning
• Rodent: Relevance determination in differential equations
• Model discovery in the sparse sampling regime
• Universal set of Observables for Forecasting Physical Systems through Causal Embedding
• Subspace Graph Physics: Real-Time Rigid Body-Driven Granular Flow Simulation
• Generative Adversarial Network for Probabilistic Forecast of Random Dynamical System