Artificial Neural Networks for Solving Ordinary and Partial Differential Equations

Abstract

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions and contains no adjustable parameters. The second part is constructed so as not to affect the boundary conditions. This part involves a feedforward neural network, containing adjustable parameters (the weights). Hence by construction the boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ODE's, to systems of coupled ODE's and also to PDE's. In this article we illustrate the method by solving a variety of model problems and present comparisons with finite elements for several cases of partial differential equations.

LAtex file, 26 pages, 21 figs, submitted to IEEE TNN

References in corpus (1)

• Artificial Neural Network Methods in Quantum Mechanics

Cited by in corpus (319)

• Machine Learning for Fluid Mechanics
• DeepXDE: A deep learning library for solving differential equations
• DGM: A deep learning algorithm for solving partial differential equations
• Adaptive activation functions accelerate convergence in deep and physics-informed neural networks
• fPINNs: Fractional Physics-Informed Neural Networks
• 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
• Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems
• Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems
• 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
• PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain
• Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
• Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons
• Weak Adversarial Networks for High-dimensional Partial Differential Equations
• Adversarial Uncertainty Quantification in Physics-Informed Neural Networks
• Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates
• Deep Learning of Vortex Induced Vibrations
• Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport
• CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method
• Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations
• Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks
• Efficient training of physics-informed neural networks via importance sampling
• Physics-informed deep learning for incompressible laminar flows
• Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
• Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture
• Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics
• Will Artificial Intelligence supersede Earth System and Climate Models?
• 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
• Physics informed neural networks for continuum micromechanics
• Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial
• Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations
• Data-assisted reduced-order modeling of extreme events in complex dynamical systems
• A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder
• PINNeik: Eikonal solution using physics-informed neural networks
• Solving parametric PDE problems with artificial neural networks
• Solving nonlinear differential equations with differentiable quantum circuits
• Super-Resolution Analysis via Machine Learning: A Survey for Fluid Flows
• Biologically-informed neural networks guide mechanistic modeling from sparse experimental data
• A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks
• Multi-Objective Loss Balancing for Physics-Informed Deep Learning
• Deep learning observables in computational fluid dynamics
• Physics Informed Neural Networks for Simulating Radiative Transfer
• Simulator-free Solution of High-Dimensional Stochastic Elliptic Partial Differential Equations using Deep Neural Networks
• Deep Learning in Deterministic Computational Mechanics
• Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
• 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
• Recent Advances and Applications of Machine Learning in Experimental Solid Mechanics: A Review
• PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries
• Data-driven rogue waves and parameter discovery in the defocusing NLS equation with a potential using the PINN deep learning
• Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations
• The Neural Particle Method -- An Updated Lagrangian Physics Informed Neural Network for Computational Fluid Dynamics
• PINN deep learning for the Chen-Lee-Liu equation: Rogue wave on the periodic background
• Distributed physics informed neural network for data-efficient solution to partial differential equations
• Deep learning of free boundary and Stefan problems
• Hamiltonian neural networks for solving equations of motion
• A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecture
• Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs
• Pricing options and computing implied volatilities using neural networks
• Physics-Informed Neural Networks for Shell Structures
• A mesh-free method for interface problems using the deep learning approach
• A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations
• Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion
• SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs
• Multilevel domain decomposition-based architectures for physics-informed neural networks
• Learning in Sinusoidal Spaces with Physics-Informed Neural Networks
• CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries
• Deep Theory of Functional Connections: A New Method for Estimating the Solutions of PDEs
• Physics-Driven Regularization of Deep Neural Networks for Enhanced Engineering Design and Analysis
• Uncovering turbulent plasma dynamics via deep learning from partial observations
• Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks
• A practical PINN framework for multi-scale problems with multi-magnitude loss terms
• Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanics
• A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems
• Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks
• Solving inverse-PDE problems with physics-aware neural networks
• Probabilistic Deep Learning for Real-Time Large Deformation Simulations
• Solving forward and inverse problems of the logarithmic nonlinear Schrodinger equation with PT-symmetric harmonic potential via deep learning
• Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis
• Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics
• Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems
• Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity
• Physics-informed neural networks for solving parametric magnetostatic problems
• Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?
• On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton--Jacobi partial differential equations
• Evolutional Deep Neural Network
• Int-Deep: A Deep Learning Initialized Iterative Method for Nonlinear Problems
• Enriched physics-informed neural networks for in-plane crack problems: Theory and MATLAB codes
• SelectNet: Self-paced Learning for High-dimensional Partial Differential Equations
• Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks
• Pre-training strategy for solving evolution equations based on physics-informed neural networks
• Uncertainty Quantification in Multivariable Regression for Material Property Prediction with Bayesian Neural Networks
• BINN: A deep learning approach for computational mechanics problems based on boundary integral equations
• Physics Informed Neural Fields for Smoke Reconstruction with Sparse Data
• A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks
• PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations
• Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh
• On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements
• Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models
• Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning
• Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method
• Enhancement of shock-capturing methods via machine learning
• Physics-informed neural networks for one-dimensional sound field predictions with parameterized sources and impedance boundaries
• A solver for subsonic flow around airfoils based on physics-informed neural networks and mesh transformation
• Using Deep Learning to Extend the Range of Air-Pollution Monitoring and Forecasting
• Efficient physics-informed neural networks using hash encoding
• Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural network
• Markov Chain Monte Carlo with Neural Network Surrogates: Application to Contaminant Source Identification
• Optimizing a DIscrete Loss (ODIL) to solve forward and inverse problems for partial differential equations using machine learning tools
• A neural network multigrid solver for the Navier-Stokes equations
• Deep learning neural networks for the third-order nonlinear Schrodinger equation: Solitons, breathers, and rogue waves
• Accelerating hydrodynamic simulations of urban drainage systems with physics-guided machine learning
• DCEM: A deep complementary energy method for solid mechanics
• Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning
• Physics-Informed Neural Networks for Time-Domain Simulations: Accuracy, Computational Cost, and Flexibility
• The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations
• Principle-driven Fiber Transmission Model based on PINN Neural Network
• Deep learning for thermal plasma simulation: solving 1-D arc model as an example
• Data vs. Physics: The Apparent Pareto Front of Physics-Informed Neural Networks
• Bayesian Physics Informed Neural Networks for Data Assimilation and Spatio-Temporal Modelling of Wildfires
• A Modified Batch Intrinsic Plasticity Method for Pre-training the Random Coefficients of Extreme Learning Machines
• Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networks
• On quadrature rules for solving Partial Differential Equations using Neural Networks
• Hyperbolic compactification of M-theory and de Sitter quantum gravity
• Reduced and All-at-Once Approaches for Model Calibration and Discovery in Computational Solid Mechanics
• Neural Network Study of Hidden-Charm Pentaquark Resonances
• A machine-learning solver for modified diffusion equations
• General solutions for nonlinear differential equations: a rule-based self-learning approach using deep reinforcement learning
• Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach
• Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized 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
• Using neural networks to accelerate the solution of the Boltzmann equation
• GINNs: Graph-Informed Neural Networks for Multiscale Physics
• Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs
• Graph Embedded Intuitionistic Fuzzy Random Vector Functional Link Neural Network for Class Imbalance Learning
• A Meshless Method to Compute Pressure Fields from Image Velocimetry
• Physics-Informed Neural Networks for Parametric Compressible Euler Equations
• Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)
• ModalPINN: an extension of Physics-Informed Neural Networks with enforced truncated Fourier decomposition for periodic flow reconstruction using a limited number of imperfect sensors
• Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networks
• On the Use of Neural Networks for Full Waveform Inversion
• PINNSim: A Simulator for Power System Dynamics based on Physics-Informed Neural Networks
• Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty
• Improved Training of Physics-Informed Neural Networks with Model Ensembles
• Meta-learning Pseudo-differential Operators with Deep Neural Networks
• Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies
• Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance
• A Neural Network Study of Blasius Equation
• 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
• Cornell Potential: A Neural Network Approach
• Robust quantum gates using smooth pulses and physics-informed neural networks
• Can Transfer Neuroevolution Tractably Solve Your Differential Equations?
• Velocity Reconstruction in Puffing Pool Fires with Physics-Informed Neural Networks
• Data-Driven Finite Elements Methods: Machine Learning Acceleration of Goal-Oriented Computations
• Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification
• Observational cosmology with Artificial Neural Networks
• A Framework of Data Assimilation for Wind Flow Fields by Physics-informed Neural Networks
• Neural Networks Enforcing Physical Symmetries in Nonlinear Dynamical Lattices: The Case Example of the Ablowitz-Ladik Model
• CEENs: Causality-enforced evolutional networks for solving time-dependent partial differential equations
• An Analysis of an Integrated Mathematical Modeling -- Artificial Neural Network Approach for the Problems with a Limited Learning Dataset
• A Levenberg-Marquardt algorithm for sparse identification of dynamical systems
• A hybrid data driven-physics constrained Gaussian process regression framework with deep kernel for uncertainty quantification
• Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks
• Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
• Physics-Informed Deep Learning of Rate-and-State Fault Friction
• A Dimension-Augmented Physics-Informed Neural Network (DaPINN) with High Level Accuracy and Efficiency
• Deep-learning quasi-particle masses from QCD equation of state
• Solving the Initial Value Problem of Ordinary Differential Equations by Lie Group based Neural Network Method
• PCNN: A physics-constrained neural network for multiphase flows
• Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs
• Enhancing approximation abilities of neural networks by training derivatives
• Physics-inspired Neural Networks for Parameter Learning of Adaptive Cruise Control Systems
• DNN-MG: A Hybrid Neural Network/Finite Element Method with Applications to 3D Simulations of the Navier-Stokes Equations
• Predicting nonlinear optical scattering with physics-driven neural networks
• Making Invisible Visible: Data-Driven Seismic Inversion with Spatio-temporally Constrained Data Augmentation
• Cosmology-informed neural networks to solve the background dynamics of the Universe
• The effect of boson-boson interaction on the Bipolaron formation
• Physics-informed Information Field Theory for Modeling Physical Systems with Uncertainty Quantification
• Reconstruction, forecasting, and stability of chaotic dynamics from partial data
• Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation
• Neural network tokamak equilibria with incompressible flows
• Constrained Hamiltonian Systems and Physics-Informed Neural Networks: Hamilton-Dirac Neural Networks
• Structure Probing Neural Network Deflation
• Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks
• Deep neural network approximations for Monte Carlo algorithms
• Imaging Conductivity from Current Density Magnitude using Neural Networks
• Solving non-linear Kolmogorov equations in large dimensions by using deep learning: a numerical comparison of discretization schemes
• opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation
• The Principle of Minimum Pressure Gradient: An Alternative Basis for Physics-Informed Learning of Incompressible Fluid Mechanics
• Machine learning based data-driven discovery of nonlinear phase-field dynamics
• SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decomposition
• Learning Free-Surface Flow with Physics-Informed Neural Networks
• A hybrid inference system for improved curvature estimation in the level-set method using machine learning
• A Physics-Informed Deep Learning Model of the Hot Tail Runaway Electron Seed
• Physics-Informed Neural Networks: a Plug and Play Integration into Power System Dynamic Simulations
• An extrapolation-driven network architecture for physics-informed deep learning
• Modelling parametric uncertainty in PDEs models via Physics-Informed Neural Networks
• Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving Linear Elasticity Problems
• Bessel Vortices in Spin-Orbit-Coupled Binary Bose-Einstein Condensates with Zeeman Splitting
• Gravitational Duals from Equations of State
• Solution of FPK Equation for Stochastic Dynamics Subjected to Additive Gaussian Noise via Deep Learning Approach
• An unsupervised deep learning approach in solving partial integro-differential equations
• The Deep Minimizing Movement Scheme
• Functional Tensor Network Solving Many-body Schrödinger Equation
• An exploratory study on machine learning to couple numerical solutions of partial differential equations
• A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method
• Solving the Pulsar Equation using Physics-Informed Neural Networks
• The Hard-Constraint PINNs for Interface Optimal Control Problems
• Physics-informed neural networks for learning the homogenized coefficients of multiscale elliptic equations
• Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients
• Local neural operator for solving transient partial differential equations on varied domains
• Partial-differential-algebraic equations of nonlinear dynamics by Physics-Informed Neural-Network: (I) Operator splitting and framework assessment
• Neural networks catching up with finite differences in solving partial differential equations in higher dimensions
• The mixed deep energy method for resolving concentration features in finite strain hyperelasticity
• Inference of neutrino flavor evolution through data assimilation and neural differential equations
• Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing
• Towards Model Reduction for Power System Transients with Physics-Informed PDE
• Neural network based approach for solving problems in plane wave duct acoustics
• A Physics-Informed Deep Learning Description of Knudsen Layer Reactivity Reduction
• Meshfree Variational Physics Informed Neural Networks (MF-VPINN): an adaptive training strategy
• Input convex neural networks: universal approximation theorem and implementation for isotropic polyconvex hyperelastic energies
• Improving Neural Network Training using Dynamic Learning Rate Schedule for PINNs and Image Classification
• Transfer Learning Enhanced Full Waveform Inversion
• Error-correcting neural networks for semi-Lagrangian advection in the level-set method
• Deep Learning for Robotic Mass Transport Cloaking
• LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometry
• Discovering Physics-Informed Neural Networks Model for Solving Partial Differential Equations through Evolutionary Computation
• Meshless physics-informed deep learning method for three-dimensional solid mechanics
• MaxwellNet: Physics-driven deep neural network training based on Maxwell's equations
• Computing non-equilibrium trajectories by a deep learning approach
• Accelerating Optimal Experimental Design for Robust Synchronization of Uncertain Kuramoto Oscillator Model Using Machine Learning
• Modeling Systems with Machine Learning based Differential Equations
• Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities
• Learning a Reduced Basis of Dynamical Systems using an Autoencoder
• Optimal Dirichlet Boundary Control by Fourier Neural Operators Applied to Nonlinear Optics
• Gravitational Models Explain Shifts on Human Visual Attention
• Artificial Neural Networks in Fluid Dynamics: A Novel Approach to the Navier-Stokes Equations
• Physics-Informed Long Short-Term Memory for Forecasting and Reconstruction of Chaos
• Physics-informed Neural Networks approach to solve the Blasius function
• Investigating Guiding Information for Adaptive Collocation Point Sampling in PINNs
• Analytical derivatives of Neural Networks
• Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?
• Physics Encoded Blocks in Residual Neural Network Architectures for Digital Twin Models
• Learning the Tangent Space of Dynamical Instabilities from Data
• Variational PINNs with tree-based integration and boundary element data in the modeling of multi-phase architected materials
• Learning Differential Operators for Interpretable Time Series Modeling
• Physics-Informed Graph-Mesh Networks for PDEs: A hybrid approach for complex problems
• Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks
• Deep learning of transition probability densities for stochastic asset models with applications in option pricing
• SPADE4: Sparsity and Delay Embedding based Forecasting of Epidemics
• Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning
• Numerical Solution and Bifurcation Analysis of Nonlinear Partial Differential Equations with Extreme Learning Machines
• Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems
• Faster Bayesian inference with neural network bundles and new results for $f(R)$ models
• CDNNs: The coupled deep neural networks for coupling of the Stokes and Darcy-Forchheimer problems
• Sequential learning based PINNs to overcome temporal domain complexities in unsteady flow past flapping wings
• Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Discontinuity Interface
• About optimal loss function for training physics-informed neural networks under respecting causality
• An efficient wavelet-based physics-informed neural network for multiscale problems
• Time-dependent Neural Galerkin Method for Quantum Dynamics
• Mass Spectrum of Exotic X(5568) State via Artificial Neural Network
• Single- to multi-fidelity history-dependent learning with uncertainty quantification and disentanglement: application to data-driven constitutive modeling
• Numerical simulation of transient heat conduction with moving heat source using Physics Informed Neural Networks
• Counting the uncounted : estimating the unaccounted COVID-19 infections in India
• Avoiding overfitting of multilayer perceptrons by training derivatives
• Neural Network Solutions to Differential Equations in Non-Convex Domains: Solving the Electric Field in the Slit-Well Microfluidic Device
• Neural Hawkes: Non-Parametric Estimation in High Dimension and Causality Analysis in Cryptocurrency Markets
• Radiative Transfer as a Bayesian Linear Regression problem
• Hamiltonian Neural Networks approach to fuzzball geodesics
• General-relativistic magnetar magnetospheres in 3D with physics-informed neural networks
• Runge-Kutta Physics Informed Neural Networks: Formulation and Analysis
• Towards a Machine-Learned Poisson Solver for Low-Temperature Plasma Simulations in Complex Geometries
• Physics-Guided Discovery of Highly Nonlinear Parametric Partial Differential Equations
• Web-enabled Intelligent System for Continuous Sensor Data Processing and Visualization
• Spectral Informed Neural Network: An Efficient and Low-Memory PINN
• Fredholm Neural Networks
• Physics-informed neural networks for solving functional renormalization group on a lattice
• Deep Ritz-Finite Element methods: Neural Network Methods trained with Finite Elements
• Linearized Implicit Methods Based on a Single-Layer Neural Network: Application to Keller-Segel Models
• Deep Learning solutions to singular ordinary differential equations: from special functions to spherical accretion
• Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities
• An approximate Riemann solver approach in Physics-Informed Neural Networks for hyperbolic conservation laws
• A novel Artificial Neural Network-based streamline tracing strategy applied to hypersonic waverider design
• A deep learning energy method for hyperelasticity and viscoelasticity
• Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks
• A deep implicit-explicit minimizing movement method for option pricing in jump-diffusion models
• Learning Trajectories of Hamiltonian Systems with Neural Networks
• Volume-preserving geometric shape optimization of the Dirichlet energy using variational neural networks
• CONFIDE: Contextual Finite Differences Modelling of PDEs
• Deep learning neural network for approaching Schrödinger problems with arbitrary two-dimensional confinement
• A Natural Deep Ritz Method for Essential Boundary Value Problems
• Understanding the surface wave characteristics using 2D particle-in-cell simulation and deep neural network
• Machine learning and serving of discrete field theories -- when artificial intelligence meets the discrete universe
• Two neural-network-based methods for solving obstacle problems
• Solving physics-based initial value problems with unsupervised machine learning
• A simple light-trapping device from a hyperbolic metamaterial on a catenoid
• Deep learning-based computational method for soft matter dynamics: Deep Onsager-Machlup method
• Formally Verified Physics-Informed Neural Control Lyapunov Functions
• Sequential Monte Carlo with Cross-validated Neural Networks for Complexity of Hyperbolic Black Hole Solutions in 4D
• Bayesian Reasoning for Physics Informed Neural Networks
• Neural semi-Lagrangian method for high-dimensional advection-diffusion problems
• Probabilistic partition of unity networks for high-dimensional regression problems
• Generalization Error Estimates of Machine Learning Methods for Solving High Dimensional Schrödinger Eigenvalue Problems
• Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs
• Prediction of acoustic field in 1-D uniform duct with varying mean flow and temperature using neural networks
• Evolution of linear matter perturbations with error-bounded bundle physics-informed neural networks
• Surrogate-data-enriched Physics-Aware Neural Networks
• Efficient Shallow Ritz Method For 1D Diffusion Problems
• Physics-informed neural networks viewpoint for solving the Dyson-Schwinger equations of quantum electrodynamics
• Stability in Training PINNs for Stiff PDEs: Why Initial Conditions Matter
• Inverse Nonlinearity Compensation of Hyperelastic Deformation in Dielectric Elastomer for Acoustic Actuation
• Generative Network-Based Reduced-Order Model for Prediction, Data Assimilation and Uncertainty Quantification
• A robust and stable hybrid neural network/finite element method for 2D flows that generalizes to different geometries
• A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems
• Jets and differential linear logic