Solving high-dimensional partial differential equations using deep learning

Abstract

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up new possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their inter-relationships.

13 pages, 6 figures

References in corpus (2)

• On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations
• Counterparty Risk Valuation: A Marked Branching Diffusion Approach

Cited by in corpus (384)

• DeepXDE: A deep learning library for solving differential equations
• Photonics for artificial intelligence and neuromorphic computing
• Surrogate Modeling for Fluid Flows Based on Physics-Constrained Deep Learning Without Simulation Data
• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks
• A unified deep artificial neural network approach to partial differential equations in complex geometries
• Integrating Machine Learning and Multiscale Modeling: Perspectives, Challenges, and Opportunities in the Biological, Biomedical, and Behavioral Sciences
• PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain
• A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate
• A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications
• Weak Adversarial Networks for High-dimensional Partial Differential Equations
• Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates
• Digital twin, physics-based model, and machine learning applied to damage detection in structures
• Flow over an espresso cup: Inferring 3D velocity and pressure fields from tomographic background oriented schlieren videos via physics-informed neural networks
• PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs
• Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios
• A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches
• Solving Many-Electron Schrödinger Equation Using Deep Neural Networks
• DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks
• A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems
• A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder
• PINNeik: Eikonal solution using physics-informed neural networks
• Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains
• Prospects and applications of photonic neural networks
• Learning Constitutive Relations using Symmetric Positive Definite Neural Networks
• Deep-learning-based surrogate flow modeling and geological parameterization for data assimilation in 3D subsurface flow
• Data-Driven Deep Learning of Partial Differential Equations in Modal Space
• Deep learning observables in computational fluid dynamics
• Tackling the Curse of Dimensionality with Physics-Informed Neural Networks
• Physics Informed Neural Networks for Simulating Radiative Transfer
• Deep Learning in Deterministic Computational Mechanics
• Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
• Physics-Informed Neural Operator for Learning Partial Differential Equations
• Solving Fokker-Planck equation using deep learning
• A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
• Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations
• Data-driven rogue waves and parameter discovery in the defocusing NLS equation with a potential using the PINN deep learning
• PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries
• A nonlocal physics-informed deep learning framework using the peridynamic differential operator
• Convergence of the Deep BSDE Method for Coupled FBSDEs
• Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions
• Deep splitting method for parabolic PDEs
• Wavefield reconstruction inversion via physics-informed neural networks
• The Modern Mathematics of Deep Learning
• When and why PINNs fail to train: A neural tangent kernel perspective
• Characterizing possible failure modes in physics-informed neural networks
• Deep learning of free boundary and Stefan problems
• Hamiltonian neural networks for solving equations of motion
• Solving Electrical Impedance Tomography with Deep Learning
• Pricing options and computing implied volatilities using neural networks
• Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations
• A mesh-free method for interface problems using the deep learning approach
• Stochastic analysis of heterogeneous porous material with modified neural architecture search (NAS) based physics-informed neural networks using transfer learning
• DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing
• SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs
• Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion
• Multi-fidelity surrogate modeling using long short-term memory networks
• Physics-Driven Regularization of Deep Neural Networks for Enhanced Engineering Design and Analysis
• An overview on deep learning-based approximation methods for partial differential equations
• Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks
• Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach
• A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
• Solving the Kolmogorov PDE by means of deep learning
• Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation
• Variational system identification of the partial differential equations governing pattern-forming physics: Inference under varying fidelity and noise
• A deep learning framework for solution and discovery in solid mechanics
• Classification of chaotic time series with deep learning
• A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems
• Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis
• Solving inverse-PDE problems with physics-aware neural networks
• Digital Twin: Values, Challenges and Enablers
• Variational Physics Informed Neural Networks: the role of quadratures and test functions
• Numerical Solution of Inverse Problems by Weak Adversarial Networks
• Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics
• On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton--Jacobi partial differential equations
• Pricing and hedging American-style options with deep learning
• SelectNet: Self-paced Learning for High-dimensional Partial Differential Equations
• Int-Deep: A Deep Learning Initialized Iterative Method for Nonlinear Problems
• Latent Space Data Assimilation by using Deep Learning
• Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks
• Graph Convolutional Neural Networks for Body Force Prediction
• Solving high-dimensional optimal stopping problems using deep learning
• Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks
• A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks
• Multi-scale Deep Neural Network (MscaleDNN) Methods for Oscillatory Stokes Flows in Complex Domains
• Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning
• On generalized residue network for deep learning of unknown dynamical systems
• Structural engineering from an inverse problems perspective
• Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems
• Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations
• Using Deep Learning to Extend the Range of Air-Pollution Monitoring and Forecasting
• Strong error analysis for stochastic gradient descent optimization algorithms
• BCR-Net: a neural network based on the nonstandard wavelet form
• In-Context Operator Learning with Data Prompts for Differential Equation Problems
• Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory
• Explaining the physics of transfer learning a data-driven subgrid-scale closure to a different turbulent flow
• Learning nonlocal constitutive models with neural networks
• Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
• Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit
• Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
• Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks
• Machine Learning from a Continuous Viewpoint
• Deep learning neural networks for the third-order nonlinear Schrodinger equation: Solitons, breathers, and rogue waves
• A rate of convergence of Physics Informed Neural Networks for the linear second order elliptic PDEs
• Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
• Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations
• A Neural Network Approach for High-Dimensional Optimal Control Applied to Multi-Agent Path Finding
• Deep learning for thermal plasma simulation: solving 1-D arc model as an example
• Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priors
• On quadrature rules for solving Partial Differential Equations using Neural Networks
• QRnet: optimal regulator design with LQR-augmented neural networks
• Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach
• A machine-learning solver for modified diffusion equations
• An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operator
• Variational training of neural network approximations of solution maps for physical models
• Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure
• Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs
• Using neural networks to accelerate the solution of the Boltzmann equation
• Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks
• Learning Pair Potentials using Differentiable Simulations
• Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations
• An augmented Lagrangian deep learning method for variational problems with essential boundary conditions
• Numerical methods for backward stochastic differential equations: A survey
• Differentiable Molecular Simulations for Control and Learning
• On the Use of Neural Networks for Full Waveform Inversion
• SIRNet: Understanding Social Distancing Measures with Hybrid Neural Network Model for COVID-19 Infectious Spread
• Learning forecasts of rare stratospheric transitions from short simulations
• Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks
• Normalizing Flows and the Real-Time Sign Problem
• Meta-learning Pseudo-differential Operators with Deep Neural Networks
• Efficient nonlinear manifold reduced order model
• Fully probabilistic deep models for forward and inverse problems in parametric PDEs
• Deep Residual Learning and PDEs on Manifold
• Learning Deep Stochastic Optimal Control Policies using Forward-Backward SDEs
• Uniform error estimates for artificial neural network approximations for heat equations
• Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
• Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions
• Deep Fictitious Play for Finding Markovian Nash Equilibrium in Multi-Agent Games
• Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networks
• Adversarial Multi-task Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations
• Data-driven discoveries of Bäcklund transforms and soliton evolution equations via deep neural network learning schemes
• A Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer
• Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problems
• Optimal Feedback Law Recovery by Gradient-Augmented Sparse Polynomial Regression
• Neural Networks Enforcing Physical Symmetries in Nonlinear Dynamical Lattices: The Case Example of the Ablowitz-Ladik Model
• Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach
• A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables
• Solving time dependent Fokker-Planck equations via temporal normalizing flow
• Physics-informed generative neural network: an application to troposphere temperature prediction
• Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Data
• Predicting rare events using neural networks and short-trajectory data
• Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks
• ChebNet: Efficient and Stable Constructions of Deep Neural Networks with Rectified Power Units via Chebyshev Approximations
• Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
• Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs
• A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver
• A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations
• Solving and Learning Nonlinear PDEs with Gaussian Processes
• Data-driven Identification of 2D Partial Differential Equations using extracted physical features
• MIM: A deep mixed residual method for solving high-order partial differential equations
• A Neural Network Approach Applied to Multi-Agent Optimal Control
• Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs
• RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators
• Stein variational gradient descent with local approximations
• RelaxNet: A structure-preserving neural network to approximate the Boltzmann collision operator
• Hypersolvers: Toward Fast Continuous-Depth Models
• A Shallow Ritz Method for Elliptic Problems with Singular Sources
• A deep learning method for solving high-order nonlinear soliton equation
• Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression
• Quantum algorithms for scientific computing
• Variational Monte Carlo - Bridging Concepts of Machine Learning and High Dimensional Partial Differential Equations
• Inverse Modeling of Viscoelasticity Materials using Physics Constrained Learning
• Structure Probing Neural Network Deflation
• Bayesian neural networks for weak solution of PDEs with uncertainty quantification
• Constrained Hamiltonian Systems and Physics-Informed Neural Networks: Hamilton-Dirac Neural Networks
• Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks
• Neural networks-based algorithms for stochastic control and PDEs in finance
• Solving non-linear Kolmogorov equations in large dimensions by using deep learning: a numerical comparison of discretization schemes
• Algorithms of Data Development For Deep Learning and Feedback Design
• Deep neural network approximations for Monte Carlo algorithms
• opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation
• Multiscale DeepONet for Nonlinear Operators in Oscillatory Function Spaces for Building Seismic Wave Responses
• Solving PDE-constrained Control Problems Using Operator Learning
• Machine learning based data-driven discovery of nonlinear phase-field dynamics
• Deep Learning Architecture Based Approach For 2D-Simulation of Microwave Plasma Interaction
• Neural Networks to solve Partial Differential Equations: a Comparison with Finite Elements
• $2^{1296}$ Exponentially Complex Quantum Many-Body Simulation via Scalable Deep Learning Method
• Four-Dimensional-Spacetime Atomistic Artificial Intelligence Models
• Towards fast weak adversarial training to solve high dimensional parabolic partial differential equations using XNODE-WAN
• Beyond Linear Response: Equivalence between Thermodynamic Geometry and Optimal Transport
• The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems
• POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition
• Pricing equity-linked life insurance contracts with multiple risk factors by neural networks
• Uncertainty quantification of two-phase flow in porous media via coupled-TgNN surrogate model
• A model-data asymptotic-preserving neural network method based on micro-macro decomposition for gray radiative transfer equations
• Deep Euler method: solving ODEs by approximating the local truncation error of the Euler method
• An unsupervised deep learning approach in solving partial integro-differential equations
• Generalised multilevel Picard approximations
• LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data
• Some observations on high-dimensional partial differential equations with Barron data
• Physics-informed attention-based neural network for solving non-linear partial differential equations
• Optimizing Variational Physics-Informed Neural Networks Using Least Squares
• On the locality of local neural operator in learning fluid dynamics
• Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients
• Towards Robust and Stable Deep Learning Algorithms for Forward Backward Stochastic Differential Equations
• Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks
• Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous media
• An Optimal Control Method to Compute the Most Likely Transition Path for Stochastic Dynamical Systems with Jumps
• Differentiable Physics: A Position Piece
• Towards a regularity theory for ReLU networks -- chain rule and global error estimates
• Deep backward schemes for high-dimensional nonlinear PDEs
• Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing
• Unsupervised Neural Networks for Quantum Eigenvalue Problems
• Neural networks catching up with finite differences in solving partial differential equations in higher dimensions
• A Priori Analysis of Stable Neural Network Solutions to Numerical PDEs
• Learning and Meta-Learning of Stochastic Advection-Diffusion-Reaction Systems from Sparse Measurements
• Solving Non-local Fokker-Planck Equations by Deep Learning
• Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations
• Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
• Enforcing exact boundary and initial conditions in the deep mixed residual method
• Deep learning based numerical approximation algorithms for stochastic partial differential equations
• Structure Preservation for the Deep Neural Network Multigrid Solver
• Reproducing Activation Function for Deep Learning
• Deep neural network approximations for the stable manifolds of the Hamilton-Jacobi-Bellman equations
• Safe Optimal Control Using Stochastic Barrier Functions and Deep Forward-Backward SDEs
• Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks
• Symbolic Equation Solving via Reinforcement Learning
• Simple lessons from complex learning: what a neural network model learns about cosmic structure formation
• Deep adaptive sampling for surrogate modeling without labeled data
• Discovery of Dynamics Using Linear Multistep Methods
• MeshingNet: A New Mesh Generation Method based on Deep Learning
• Enforcing constraints for time series prediction in supervised, unsupervised and reinforcement learning
• Classification-based detection and quantification of cross-domain data bias in materials discovery
• Risk management with machine-learning-based algorithms
• Monte Carlo for high-dimensional degenerated Semi Linear and Full Non Linear PDEs
• Deciphering and integrating invariants for neural operator learning with various physical mechanisms
• Deep Learning-Based BSDE Solver for Libor Market Model with Application to Bermudan Swaption Pricing and Hedging
• Meta-MgNet: Meta Multigrid Networks for Solving Parameterized Partial Differential Equations
• Market making and incentives design in the presence of a dark pool: a deep reinforcement learning approach
• Meshless physics-informed deep learning method for three-dimensional solid mechanics
• Accelerating Optimal Experimental Design for Robust Synchronization of Uncertain Kuramoto Oscillator Model Using Machine Learning
• Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers
• NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations
• A deep surrogate approach to efficient Bayesian inversion in PDE and integral equation models
• Deep learning methods for the computation of vibrational wavefunctions
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation
• Deep 2FBSDEs For Systems With Control Multiplicative Noise
• Spacetime Neural Network for High Dimensional Quantum Dynamics
• On the Representation of Solutions to Elliptic PDEs in Barron Spaces
• Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures
• Learning Differential Operators for Interpretable Time Series Modeling
• An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions
• Solving PDEs on Unknown Manifolds with Machine Learning
• Neural network optimal feedback control with enhanced closed loop stability
• Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling
• Error Approximation and Bias Correction in Dynamic Problems using a Recurrent Neural Network/Finite Element Hybrid Model
• Physics-Informed Neural Networks for Nonlocal Beam Eigenvalue Problems
• Partition of unity networks: deep hp-approximation
• Numerical Solution and Bifurcation Analysis of Nonlinear Partial Differential Equations with Extreme Learning Machines
• Unbiased deep solvers for linear parametric PDEs
• Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality
• Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning
• Learning phase field mean curvature flows with neural networks
• Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory
• Optimal Policies for a Pandemic: A Stochastic Game Approach and a Deep Learning Algorithm
• DOSnet as a Non-Black-Box PDE Solver: When Deep Learning Meets Operator Splitting
• Neural Network Solutions to Differential Equations in Non-Convex Domains: Solving the Electric Field in the Slit-Well Microfluidic Device
• Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning
• Parametric Complexity Bounds for Approximating PDEs with Neural Networks
• High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations
• Feature Engineering with Regularity Structures
• MCMC-Net: Accelerating Markov Chain Monte Carlo with Neural Networks for Inverse Problems
• Machine-learning-based methods for output only structural modal identification
• Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networks
• Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning
• A deep neural network approach on solving the linear transport model under diffusive scaling
• A Neural Solver for Variational Problems on CAD Geometries with Application to Electric Machine Simulation
• Unsupervised Reservoir Computing for Solving Ordinary Differential Equations
• Exploring complex pattern formation with convolutional neural networks
• Implicit bias with Ritz-Galerkin method in understanding deep learning for solving PDEs
• The Power of Contrast for Feature Learning: A Theoretical Analysis
• Three algorithms for solving high-dimensional fully-coupled FBSDEs through deep learning
• Epidemic control via stochastic optimal control
• Solving path dependent PDEs with LSTM networks and path signatures
• Backward Deep BSDE Methods and Applications to Nonlinear Problems
• Web-enabled Intelligent System for Continuous Sensor Data Processing and Visualization
• Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control
• Physics-Guided Discovery of Highly Nonlinear Parametric Partial Differential Equations
• Lax-Oleinik-type formulas and efficient algorithms for certain high-dimensional optimal control problems
• POMDPs in Continuous Time and Discrete Spaces
• Fredholm Neural Networks
• Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging
• Actor-Critic Algorithm for High-dimensional Partial Differential Equations
• Analysis of Deep Ritz Methods for Laplace Equations with Dirichlet Boundary Conditions
• Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities
• NOVAS: Non-convex Optimization via Adaptive Stochastic Search for End-to-End Learning and Control
• Partial Differential Equations is All You Need for Generating Neural Architectures -- A Theory for Physical Artificial Intelligence Systems
• Strong solutions of forward-backward stochastic differential equations with measurable coefficients
• Physics-informed neural networks for solving functional renormalization group on a lattice
• Spectral Informed Neural Network: An Efficient and Low-Memory PINN
• Physics-Informed Neural Networks for Solving Multiscale Mode-Resolved Phonon Boltzmann Transport Equation
• Interpolation between Residual and Non-Residual Networks
• A deep implicit-explicit minimizing movement method for option pricing in jump-diffusion models
• Higher-order Quasi-Monte Carlo Training of Deep Neural Networks
• Discovery of Governing Equations with Recursive Deep Neural Networks
• Deep learning for quadratic hedging in incomplete jump market
• Learning optimal multigrid smoothers via neural networks
• Portfolio Optimization with Feedback Strategies Based on Artificial Neural Networks
• Studying First Passage Problems using Neural Networks: A Case Study in the Slit-Well Microfluidic Device
• The Seven-League Scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations
• A Low Rank Neural Representation of Entropy Solutions
• Deep Learning-Based Least Square Forward-Backward Stochastic Differential Equation Solver for High-Dimensional Derivative Pricing
• Numerical approximation of singular Forward-Backward SDEs
• Solving continuum and rarefied flows using differentiable programming
• Spatiotemporal wall pressure forecast of a rectangular cylinder with physics-aware DeepU-Fourier neural network
• Generalization Error Analysis of Neural networks with Gradient Based Regularization
• CONFIDE: Contextual Finite Differences Modelling of PDEs
• Data-driven Hedging of Stock Index Options via Deep Learning
• Novel multi-step predictor-corrector schemes for backward stochastic differential equations
• Deep learning of material transport in complex neurite networks
• On the approximation of functions by tanh neural networks
• Methods to Recover Unknown Processes in Partial Differential Equations Using Data
• Error bounds for PDE-regularized learning
• Integration of Fractional Order Black-Scholes Merton with Neural Network
• Markov-Modulated Affine Processes
• Solving physics-based initial value problems with unsupervised machine learning
• Extraction of nonlinearity in neural networks with Koopman operator
• Machine learning moment closure models for the radiative transfer equation III: enforcing hyperbolicity and physical characteristic speeds
• Numerical computation of probabilities for nonlinear SDEs in high dimension using Kolmogorov equation
• Adaptive Learning on the Grids for Elliptic Hemivariational Inequalities
• Deep learning of parameterized equations with applications to uncertainty quantification
• Learning the GENERIC evolution
• Understanding ML driven HPC: Applications and Infrastructure
• Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations
• A physics-aware deep learning model for shear band formation around collapsing pores in shocked reactive materials
• A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs
• Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations
• Solving Backward Doubly Stochastic Differential Equations through Splitting Schemes
• On the numerical integration of the Fokker-Planck equation driven by a mechanical force and the Bismut-Elworthy-Li formula
• Approximation of Functionals by Neural Network without Curse of Dimensionality
• Embedding Principle in Depth for the Loss Landscape Analysis of Deep Neural Networks
• On Calibration Neural Networks for extracting implied information from American options
• An efficient stochastic particle method for high-dimensional nonlinear PDEs
• Natural Model Reduction for Kinetic Equations
• Probabilistic partition of unity networks for high-dimensional regression problems
• Discovering Phase Field Models from Image Data with the Pseudo-spectral Physics Informed Neural Networks
• Turbulence closure modeling with data-driven techniques: Investigation of generalizable deep neural networks
• Committor functions via tensor networks
• Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks
• Deep neural network for optimal retirement consumption in defined contribution pension system
• Learning the Dynamics of Sparsely Observed Interacting Systems
• Simultaneous upper and lower bounds of American-style option prices with hedging via neural networks
• Deep Residual Networks Learn the Geodesic Curve in the Wasserstein Space
• Adaptive feature capture method for solving partial differential equations with near singular solutions
• Generalization Error Estimates of Machine Learning Methods for Solving High Dimensional Schrödinger Eigenvalue Problems
• Error Estimation and Correction from within Neural Network Differential Equation Solvers
• Review: Ordinary Differential Equations For Deep Learning
• Design of A Two-point Steering Path Planner Using Geometric Control
• Neural Networks in Fréchet spaces
• Solving the Reaction-Diffusion equation based on analytical methods and deep learning algorithm; the Case study of sulfate attack to concrete
• Risk-indifference Pricing of American-style Contingent Claims
• Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems
• Neural Delay Differential Equations
• Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs
• Estimating Adsorption Isotherm Parameters in Chromatography via a Virtual Injection Promoting Feed-forward Neural Network
• (Thesis) Reservoir Computing With Dynamical Systems
• Solving stochastic optimal control problem via stochastic maximum principle with deep learning method
• Large-Scale Multi-Agent Deep FBSDEs
• Learning To Solve Differential Equations Across Initial Conditions
• Stationary Density Estimation of Itô Diffusions Using Deep Learning
• PDGM: a Neural Network Approach to Solve Path-Dependent Partial Differential Equations
• Finite Difference Nets: A Deep Recurrent Framework for Solving Evolution PDEs
• Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations
• Polynomial-Spline Neural Networks with Exact Integrals
• On the space-time expressivity of ResNets
• State Constrained Stochastic Optimal Control Using LSTMs
• Solving time-dependent parametric PDEs by multiclass classification-based reduced order model
• A deep learning algorithm for optimal investment strategies
• Deep Operator BSDE: a Numerical Scheme to Approximate Solution Operators
• Solving Image PDEs with a Shallow Network
• XNet-Enhanced Deep BSDE Method and Numerical Analysis
• Deep learning for gradient flows using the Brezis-Ekeland principle
• Predicting Stress in Two-phase Random Materials and Super-Resolution Method for Stress Images by Embedding Physical Information
• Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation