Bayesian PINNs solve noisy PDE problems with physics constraints.
problem Uncertainty quantification in noisy PDE problems.
method Bayesian framework combining PINNs and HMC/VI for posterior estimation.
result HMC outperforms VI for noisy data.
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48 results for “Bayesian PINN”
Bayesian PINNs tackle uncertainty in inverse problems.
problem Uncertainty quantification in inverse problems.
method Hierarchical Bayesian formulation of PINNs with variational inference and Monte Carlo dropout.
result Quantification of uncertainties in reconstructed images.
Functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networksphysics.geo-ph
Unified framework for Bayesian PDE-constrained inversion using physics-informed neural networks.
problem Incorporating prior distributions in function space into Bayesian PINN-based inversion.
method Functional-prior-based approaches (fpBPINN) to Bayesian PDE-constrained inversion using physics-informed neural networks (PINNs). Two complementary approaches: FPI-BPINN and fParVI-PINN.
result Accurate estimation of posterior distributions in seismic traveltime tomography and Darcy-flow permeability inversion.
Bayesian PINN improves estimation of PDE solutions from noisy data.
problem Estimating solutions of PDEs from noisy measurements.
method Bayesian approach to Physics-informed neural networks (PINNs) for inverse problems.
result Convergence rate of Bayesian posterior mean error in PDE solutions.
Paper improves uncertainty quantification in PINNs using error bounds and solution bundles.
problem Uncertainty quantification in PINNs for differential equation systems.
method Two-step procedure with Bayesian Neural Networks and heteroscedastic variance.
result Improved uncertainty estimation over PINNs solutions in differential equation systems.
Bayesian PINNs learn elliptic PDEs with near-minimax posterior contraction rate.
problem Learning elliptic PDEs with noisy data and non-homogeneous boundary conditions.
method Bayesian approach with a Hölder space prior on neural network weights.
result Posterior contracts at near-minimax rate without prior knowledge of solution smoothness.
Repulsive ensembles improve uncertainty estimates in PINNs for differential equations.
problem Improving uncertainty estimates in PINNs for differential equations.
method Employing repulsive ensembles (RE-PINN) with a repulsive term in the loss function.
result Repulsive ensembles produce more accurate uncertainty estimates and higher sample diversity.
Paper discovers differential equations from data using neural networks and Bayesian methods.
problem Discovering differential equations from datasets using machine learning.
method Integrates neural network-based surrogates with Sparse Bayesian Learning (SBL).
result Proposes a robust model discovery algorithm and a Physics Informed Normalizing Flow (PINF).
Unified Bayesian PINN framework for solving inverse problems in infrared image processing.
problem Solving inverse problems in high-dimensional settings with complex physics.
method Bayesian Physics-Informed Neural Networks (BPINN-IP) framework, incorporating physical laws and uncertainties.
result Unified framework for physical constraints, prior knowledge, and data-driven inference with uncertainty quantification.
New method uses EKI for efficient Bayesian inference in high-dimensional problems.
problem Efficient inference for high-dimensional posterior distributions in physics-informed neural networks.
method Ensemble Kalman Inversion (EKI) for high-dimensional posterior inference.
result EKI-based inference provides comparable uncertainty estimates to HMC-based methods but with reduced computational cost.
Bayesian Reasoning for Physics Informed Neural Networksphysics.comp-ph
Bayesian PINNs optimize loss weights for PDEs and data.
problem Optimizing loss weights in physics-informed neural networks.
method Laplace approximation for efficient model evidence computation.
result Unified Bayesian setting for PDEs and noisy measurements.
Uncertainty Quantification for Physics-Informed Neural Networks with Extended Fiducial Inferencestat.ML
New method for PINNs uncertainty quantification without prior distribution.
problem Lack of reliable uncertainty quantification for PINNs.
method Extended fiducial inference with narrow-neck hyper-network.
result Construction of honest confidence sets based on observed data.
A Physics Informed Machine Learning Framework for Optimal Sensor Placement and Parameter Estimationstat.ML
Study optimizes sensor placement for accurate parameter estimation in complex systems.
problem Challenges in parameter estimation with limited or noisy data.
method Physics-Informed Neural Networks (PINNs) for optimal sensor placement and parameter estimation.
result PINNs-based framework achieves higher accuracy in parameter estimation compared to random sensor placements.
PINN-FEM combines PINNs and FEM for accurate Dirichlet boundary condition enforcement.
problem Challenges in enforcing Dirichlet boundary conditions in PINNs.
method Hybrid approach combining PINNs and FEM for strong boundary condition enforcement.
result PINN-FEM outperforms standard PINN models in accuracy and robustness.
PD-PINNs accelerate PINN training by incorporating task-specific dictionaries.
problem Training PINNs is slow and lacks theoretical error bounds.
method Integrates task-dependent dictionaries into PINNs to enhance convergence.
result PD-PINNs achieve faster convergence and bounded prediction errors.
Paper explores challenges in training PINNs and loss landscape effects.
problem Challenges in training Physics-Informed Neural Networks (PINNs) due to loss landscape issues.
method Examined gradient-based optimizers Adam, L-BFGS, and their combination Adam+L-BFGS, and introduced NysNewton-CG (NNCG).
result Adam+L-BFGS outperforms other optimizers, and NysNewton-CG significantly improves PINN performance.
Developed MF-PINNs to solve coupled Stokes-Darcy equations more accurately.
problem Solving coupled Stokes-Darcy equations with varying physical constants.
method Combining VP and SV forms with adjusted weights in MF-PINNs.
result Improved accuracy of streamline and pressure fields in numerical experiments.
Adaptive PINNs improve accuracy by adding points where solutions are uncertain.
problem Inadequate sampling in PINNs leads to inaccurate solutions, especially near singularities.
method FI-PINNs use failure probability to dynamically add points, improving numerical accuracy.
result FI-PINNs achieve better accuracy through adaptive sampling, as proven by rigorous error bounds.
New method improves training of PINNs for PDEs by adding noisy supervision terms.
problem Slow or failed convergence of PINNs on challenging PDEs.
method Operator preconditioning using Feynman-Kac supervision and non-asymptotic error bounds.
result Non-asymptotic error bounds for FK-PINNs, showing improved performance over standard PINNs.
SGPT-PINNs solve PDEs with sparse, small models.
problem Efficiently solving parametric PDEs with minimal resources.
method Sparse and small architecture, mathematically rigorous greedy algorithm, knowledge distillation, down-sampling.
result Achieves high efficiency with significantly fewer parameters.
New PINN formulation respects causality for complex systems.
problem Existing PINNs fail to accurately simulate chaotic systems.
method Proposed a simple re-formulation of PINNs loss functions to respect physical causality.
result Significant accuracy improvements across chaotic systems.
PINNs solve differential geometry problems in complex shapes.
problem Solving differential geometry problems in complex shapes.
method Training neural networks with loss functions inspired by differential conditions.
result PINNs are effective for differential geometry problems.
This work analyzes PINNs for advection-diffusion equations using NTK theory.
problem Understanding and resolving the training difficulties of PINNs for advection-diffusion equations.
method Neural Tangent Kernel (NTK) analysis of PINNs for the linear advection-diffusion equation (LAD).
result PINNs struggle due to spectral bias and convergence rate disparity, especially in advection-dominated and diffusion-dominated regimes.
PINNACLE optimizes point selection for PINNs, improving accuracy.
problem Challenges in selecting points for training Physics-Informed Neural Networks (PINNs).
method Introduces PINNACLE, an algorithm that jointly optimizes collocation and experimental points selection, adjusting point proportions dynamically.
result PINNACLE outperforms existing methods in forward, inverse, and transfer learning problems.
Koopman-PINN framework improves epidemic model parameter inference and forecasting
problem Epidemic model parameter inference and forecasting
method Combining Koopman operator theory and physics-informed learning
result More accurate parameter estimation, trajectory reconstruction, and long-term forecasting
The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial differential equations (PDEs). However, the literature lacks detailed investigat…
PINNs struggle with data-to-PDE inconsistencies, limiting their accuracy.
problem Data inconsistency in PINNs affects their accuracy and convergence.
method Systematic analysis of PINNs with varying data fidelity and residual errors.
result PINNs saturate at an error level dictated by data inconsistency.
Paper improves training physics-informed neural networks with model ensembles.
problem Training physics-informed neural networks (PINNs) is difficult due to convergence to wrong solutions.
method Proposes training an ensemble of PINNs, using ensemble agreement to expand the solution interval.
result Algorithm stabilizes PINN training and yields competitive performance.
Diffusion-PINN Samplerstat.ML
DPS uses PINNs to estimate drift in diffusion models for sampling.
problem Accurately estimating drift term in reverse SDE from unnormalized density.
method Diffusion-PINN Sampler (DPS) solves PINN for log-density of SDE marginals.
result DPS achieves convergence guarantees and accurately samples complex distributions.
Self-adaptive PINNs improve accuracy in stiff PDEs.
problem Accurate numerical solution of stiff PDEs using neural networks.
method Adaptive weights applied to each training point individually, using a soft attention mechanism.
result SA-PINNs outperform other PINN algorithms in L2 error with fewer training epochs.
Physics-Informed Neural Networks (PINNs) benchmarked against clinical estimator and reveal parameter identifiability
problem Chemotherapy pharmacokinetics (PK) with tissue concentration not measured
method Physics-Informed Neural Networks (PINNs) benchmarked against clinical estimator and reveal parameter identifiability
result PINN recovers tissue concentration and identifies non-identifiable parameters
Paper proposes a method to verify PINN fidelity using Fisher information from dynamical systems.
problem Quantifying PINN fidelity beyond simple trajectory prediction.
method Employing Fisher information for differentiable dynamical systems to compare PINN's learned equations with analytical models.
result PINN fidelity is verified by matching Fisher information landscapes of learned equations and analytical models.
LPINNs solve complex PDEs by reformulating PINNs on Lagrangian frame, reducing training complexity.
problem Complexity in training PINNs, especially for convection-diffusion equations.
method Propose LPINNs, a Lagrangian reformulation of PINNs, with two branches solving state variables and characteristics curves.
result Loss landscapes of LPINNs are less sensitive to problem complexity compared to traditional PINNs.
Paper improves PINNs' extrapolation by TL and adaptive AFs.
problem PINNs' poor extrapolation performance and sensitivity to AFs.
method Transfer learning within an extended domain and adaptive activation functions.
result Average 40% reduction in relative L2 error and 50% in mean absolute error in extrapolation domain.
PINNs can learn trivial solutions; new approach improves performance.
problem PINNs fail to learn non-trivial solutions in data-scarce settings.
method Developed an alternative sampling approach and new penalty term.
result PINNs can be improved to learn non-trivial solutions with up to 80% fewer collocation points.
The paper examines the neural tangent kernel for PINNs solving general PDEs and finds convergence conditions.
problem Analyzing the convergence of neural tangent kernel for PINNs solving general PDEs.
method Analysis of NTK initialization and convergence during training for general PDEs using PINNs.
result Homogeneity of differential operators is crucial for NTK convergence.
This study evaluates the importance of design of experiments for PINN in physics-informed deep learning.
problem Accuracy of PINN predictions depends on the design of experiment scheme.
method Comparative study of five PDEs using different design of experiment schemes.
result Hammersley sampling-based PINN outperforms other design of experiment schemes.
RS-PINN uses randomized smoothing to speed up high-dimensional PDE simulations without sacrificing accuracy.
problem High computational cost and bias in PINNs for high-dimensional PDEs.
method Introduces Gaussian noise for stochastic smoothing of PINNs, enabling Monte Carlo derivative approximation.
result Proposes bias correction techniques and a hybrid method to optimize the bias-variance trade-off.
Develops weak PINNs for efficient manifold solutions of hyperbolic equations.
problem Challenges in approximating weak solutions of nonlinear hyperbolic equations on manifolds.
method Introduces a novel weak PINN (wPINN) formulation on manifolds leveraging well-posedness theory.
result Demonstrates efficient approximation of entropy solutions on manifolds with a complexity independent of ambient space dimension.
Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiat…
DCGD improves training of PINNs by adjusting gradients to avoid negative inner products.
problem Pathological behaviors in PINNs training, especially gradient imbalance.
method Dual Cone Gradient Descent (DCGD) framework to adjust gradient direction.
result DCGD outperforms other optimization algorithms in various evaluation metrics.
Paper improves robustness of PINNs by smoothing and quantifying uncertainty.
problem PINNs can overfit and converge to local minima due to training data errors.
method Introduces Gaussian Process smoothing and sparse inducing points.
result Recover robust PINN performance against noise and errors.
A nonlocal physics-informed deep learning framework using the peridynamic differential operatorcs.LG
Develops a nonlocal PINN framework using PDDO for better solution of PDEs with sharp gradients.
problem Dealing with sharp gradients in solutions of PDEs using traditional PINN approaches.
method Integrates long-range interactions (nonlocality) into PINN using Peridynamic Differential Operator (PDDO).
result Nonlocal PINN approach improves solution accuracy and parameter inference for problems with sharp gradients.
The paper compares PINN methods for solving drift-diffusion equations on metric graphs.
problem Solving drift-diffusion equations on metric graphs using machine learning.
method Comparison of physics-informed neural networks (PINNs) for solving drift-diffusion equations on metric graphs.
result PINNs offer a flexible and versatile tool for solving parameter identification or optimization problems on metric graphs.
New framework establishes positivity of DNTK for PINNs.
problem Establishing positivity of NTK for PINNs with multiple differential operators.
method Proposed Differential Neural Tangent Kernel (DNTK) for PINNs.
result Positivity of infinite width DNTK for various activation functions and differential operators.
New framework explains neural network bias in solving differential equations.
problem Understanding and controlling the bias in PINNs for differential equations.
method Deriving an integro-differential equation from PINNs and GPR equivalence.
result PINN predictions are influenced by a kernel term reflecting architecture choices.
Physics-informed neural networks simulate radiative transfer efficiently.
problem Simulating radiative transfer accurately and efficiently.
method Physics-informed neural networks trained to minimize radiative transfer equations.
result PINNs provide an easy-to-implement, robust, and accurate method for radiative transfer simulation.
New method uses dynamic sampling to improve PINNs efficiency.
problem Improving sample efficiency and performance of PINNs.
method pdPINN, inspired by Eulerian formulation, uses dynamic Monte Carlo sampling from particle positions.
result Higher sample efficiency and improved performance of PINNs.