Fourier Neural Operator parameterizes integral kernels in Fourier space to learn parametric PDE solution operators, delivering up to 1000x speedups and zero-shot super-resolution on turbulent Navier-Stokes flows.
The Deep Ritz Method: A Deep Learning-Based Numerical Algo- rithm for Solving Variational Problems
9 Pith papers cite this work, alongside 1,349 external citations. Polarity classification is still indexing.
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A weighted FOSLS formulation for deep neural networks solves transmission problems robustly, with proofs that the loss aligns with the energy norm independently of material contrast and shows passive variance reduction.
Shallow neural networks with time-frequency localized activations achieve dimension-independent Sobolev approximation rates of order N^{-1/2} for functions in weighted modulation spaces.
Graph Kernel Networks learn PDE solution operators that generalize across discretization methods and grid resolutions using graph-based kernel integration.
SplineNet constructs a neural network whose architecture exactly reproduces isogeometric spline basis functions via Bézier extraction, enabling both data-free PDE solving and operator learning for complex shell structures.
SS-POD augments standard POD-Galerkin with a spectral-subspace partition and local POD to achieve lower out-of-sample error than either plain POD or pure spectral-Galerkin when only a handful of snapshots are available.
Bio-PINNs with a near-to-far curriculum and deformation-uncertainty proxy recover cell-induced densified phases and tether morphologies more reliably than standard adaptive PINN baselines in single-cell and multicellular settings.
Systematic benchmark of PINN architectures on 1D stiff PNP system finds BRDR loss weighting competitive with NTK at lower wall-clock time.
PINNs for first-order plane-strain elastodynamics achieve higher accuracy with soft boundary enforcement over implicit geometries but require longer training than hard enforcement.
citing papers explorer
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Fourier Neural Operator for Parametric Partial Differential Equations
Fourier Neural Operator parameterizes integral kernels in Fourier space to learn parametric PDE solution operators, delivering up to 1000x speedups and zero-shot super-resolution on turbulent Navier-Stokes flows.
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Robust Deep FOSLS for Transmission Problems
A weighted FOSLS formulation for deep neural networks solves transmission problems robustly, with proofs that the loss aligns with the energy norm independently of material contrast and shows passive variance reduction.
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Time-Frequency Analysis for Neural Networks
Shallow neural networks with time-frequency localized activations achieve dimension-independent Sobolev approximation rates of order N^{-1/2} for functions in weighted modulation spaces.
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Neural Operator: Graph Kernel Network for Partial Differential Equations
Graph Kernel Networks learn PDE solution operators that generalize across discretization methods and grid resolutions using graph-based kernel integration.
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SplineNet: An Isogeometric Deep Learning Method for Complex Shells
SplineNet constructs a neural network whose architecture exactly reproduces isogeometric spline basis functions via Bézier extraction, enabling both data-free PDE solving and operator learning for complex shell structures.
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A spectral-subspace-augmented POD-Galerkin method for parametrized PDEs with limited snapshot data
SS-POD augments standard POD-Galerkin with a spectral-subspace partition and local POD to achieve lower out-of-sample error than either plain POD or pure spectral-Galerkin when only a handful of snapshots are available.
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Cell-induced densification and tether formation in fibrous extracellular matrices with biomimetic physics-informed neural networks
Bio-PINNs with a near-to-far curriculum and deformation-uncertainty proxy recover cell-induced densified phases and tether morphologies more reliably than standard adaptive PINN baselines in single-cell and multicellular settings.
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A Systematic Benchmark of Physics-Informed Neural Network Architectures for the Stiff Poisson-Nernst-Planck System: Adaptive LossWeighting and Multi-Scale Resolution
Systematic benchmark of PINN architectures on 1D stiff PNP system finds BRDR loss weighting competitive with NTK at lower wall-clock time.
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Exact Boundary Enforcement Along Implicit Geometries for Physics-Informed, Deep Learning Problems in Continuum Mechanics
PINNs for first-order plane-strain elastodynamics achieve higher accuracy with soft boundary enforcement over implicit geometries but require longer training than hard enforcement.