DeepONet learns nonlinear operators for differential equations via branch and trunk sub-networks, achieving high-order error convergence on small datasets.
Multistep Neural Networks for Data -driven Discovery of Nonlinear Dynamical Systems
9 Pith papers cite this work. Polarity classification is still indexing.
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abstract
The process of transforming observed data into predictive mathematical models of the physical world has always been paramount in science and engineering. Although data is currently being collected at an ever-increasing pace, devising meaningful models out of such observations in an automated fashion still remains an open problem. In this work, we put forth a machine learning approach for identifying nonlinear dynamical systems from data. Specifically, we blend classical tools from numerical analysis, namely the multi-step time-stepping schemes, with powerful nonlinear function approximators, namely deep neural networks, to distill the mechanisms that govern the evolution of a given data-set. We test the effectiveness of our approach for several benchmark problems involving the identification of complex, nonlinear and chaotic dynamics, and we demonstrate how this allows us to accurately learn the dynamics, forecast future states, and identify basins of attraction. In particular, we study the Lorenz system, the fluid flow behind a cylinder, the Hopf bifurcation, and the Glycoltic oscillator model as an example of complicated nonlinear dynamics typical of biological systems.
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Neural networks are redefined as continuous dynamical systems by learning the derivative of the hidden state with a neural network and integrating it with an ODE solver.
Numerical integration schemes used in optimizing models of dynamical systems from sampled data can distort learned stability properties, inducing anti-damping artifacts in originally damped systems.
Universal Differential Equations unify scientific models with machine learning by embedding flexible approximators into differential equations, enabling applications from biological mechanism discovery to high-dimensional optimization.
PnP-Corrector decouples physics simulation from error correction via a plug-and-play agent, cutting error by 29% in 300-day global ocean-atmosphere forecasts.
A quantum ensemble method reduces operator inference to linear complexity and supplies distribution-free uncertainty bounds for high-dimensional dynamical systems.
Dynamics-encoded deep learning approaches are developed for system identification and parameter estimation in dynamical systems using numerical discretization schemes.
μ-FlowNet applies an attention U-Net to map flow fields in irregular microchannels, reporting dice score 0.9317 and IoU 0.8731 on test data while outperforming standard U-Net and T-Net.
A physics-informed neural network merges sparse LBM data with Navier-Stokes equations to predict unsteady flows in fractal-rough microchannels at 150-200 times lower data cost.
citing papers explorer
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DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
DeepONet learns nonlinear operators for differential equations via branch and trunk sub-networks, achieving high-order error convergence on small datasets.
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Neural Ordinary Differential Equations
Neural networks are redefined as continuous dynamical systems by learning the derivative of the hidden state with a neural network and integrating it with an ODE solver.
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Artifacts of Numerical Integration in Learning Dynamical Systems
Numerical integration schemes used in optimizing models of dynamical systems from sampled data can distort learned stability properties, inducing anti-damping artifacts in originally damped systems.
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Universal Differential Equations for Scientific Machine Learning
Universal Differential Equations unify scientific models with machine learning by embedding flexible approximators into differential equations, enabling applications from biological mechanism discovery to high-dimensional optimization.
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PnP-Corrector: A Universal Correction Framework for Coupled Spatiotemporal Forecasting
PnP-Corrector decouples physics simulation from error correction via a plug-and-play agent, cutting error by 29% in 300-day global ocean-atmosphere forecasts.
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Conformalized Quantum DeepONet Ensembles for Scalable Operator Learning with Distribution-Free Uncertainty
A quantum ensemble method reduces operator inference to linear complexity and supplies distribution-free uncertainty bounds for high-dimensional dynamical systems.
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Dynamics-Encoded Deep Learning for Robust System Identification and Parameter Estimation
Dynamics-encoded deep learning approaches are developed for system identification and parameter estimation in dynamical systems using numerical discretization schemes.
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$\mu$-FlowNet: A Deep Learning Approach for Mapping Flow Fields in Irregular Microchannels Using an Attention-based U-Net Encoder-Decoder Architecture
μ-FlowNet applies an attention U-Net to map flow fields in irregular microchannels, reporting dice score 0.9317 and IoU 0.8731 on test data while outperforming standard U-Net and T-Net.
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Amalgamation of Physics-Informed Neural Network and LBM for the Prediction of Unsteady Fluid Flows in Fractal-Rough Microchannels
A physics-informed neural network merges sparse LBM data with Navier-Stokes equations to predict unsteady flows in fractal-rough microchannels at 150-200 times lower data cost.