Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning | Paper Citation Record

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This paper cites PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks

Reference 1

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This paper cites Neural operators for accelerating scientific simulations and design.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Neural operators for accelerating scientific simulations and design

Reference 2

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Layer Normalization

Reference 4

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This paper cites Automatic differentiation in machine learning: a survey.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Automatic differentiation in machine learning: a survey

Reference 5

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning A Foundation Model for the Earth System

Reference 6

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning The mathematical theory of finite element methods

Reference 7

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This paper cites The Hadamard- PINN for PDE inverse problems: Convergence with distant initial guesses.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning The Hadamard- PINN for PDE inverse problems: Convergence with distant initial guesses

Reference 8

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Neural ordinary differential equations

Reference 9

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This paper cites CAN-PINN: A fast physics-informed neural network based on coupled- automatic–numerical differentiation method.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning CAN-PINN: A fast physics-informed neural network based on coupled- automatic–numerical differentiation method

Reference 10

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This paper cites Separable physics-informed neural networks.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Separable physics-informed neural networks

Reference 11

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This paper cites An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale

Reference 12

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This paper cites Reaction–diffusion processes at the nano-and microscales.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Reaction–diffusion processes at the nano-and microscales

Reference 13

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This paper cites Automatic differentiation is no panacea for phylogenetic gradient computation.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Automatic differentiation is no panacea for phylogenetic gradient computation

Reference 14

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This paper cites The Identity of Weak and Strong Extensions of Differential Operators.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning The Identity of Weak and Strong Extensions of Differential Operators

Reference 15

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This paper cites Limitations of physics informed machine learning for nonlinear two-phase transport in porous media.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Limitations of physics informed machine learning for nonlinear two-phase transport in porous media

Reference 16

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This paper cites Chromatin organization in the mam- malian nucleus.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Chromatin organization in the mam- malian nucleus

Reference 17

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Observation 75713d2f-76e4-41ae-b98f-49c14a7a2004 · outbound

This paper cites Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems

Reference 18

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This paper cites Transformer Meets Boundary Value Inverse Problems.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Transformer Meets Boundary Value Inverse Problems

Reference 19

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Deep residual learning for image recognition

Reference 20

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This paper cites Aberrant chromatin reorganization in cells from diseased fibrous connective tissue in response to altered chemomechanical cues.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Aberrant chromatin reorganization in cells from diseased fibrous connective tissue in response to altered chemomechanical cues

Reference 21

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This paper cites Parameter estimation and uncertainty analysis in hydrological modeling.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Parameter estimation and uncertainty analysis in hydrological modeling

Reference 22

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Efficient physics-informed neural networks using hash encoding

Reference 23

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This paper cites The finite element method: linear static and dynamic finite element analysis.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning The finite element method: linear static and dynamic finite element analysis

Reference 24

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This paper cites Statistical and computational inverse problems.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Statistical and computational inverse problems

Reference 25

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This paper cites Active transcription and epigenetic reactions synergistically regulate meso-scale genomic organization.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Active transcription and epigenetic reactions synergistically regulate meso-scale genomic organization

Reference 26

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This paper cites Physics-informed machine learning.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Physics-informed machine learning

Reference 27

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Adam: A Method for Stochastic Optimization

Reference 28

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This paper cites Neural Green’s Function Accelerated Iterative Methods for Solving Indefinite Boundary Value Problems.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Neural Green’s Function Accelerated Iterative Methods for Solving Indefinite Boundary Value Problems

Reference 29

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This paper cites Fourier Neural Operator for Parametric Partial Differential Equations.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Fourier Neural Operator for Parametric Partial Differential Equations

Reference 30

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Physics-informed neural operator for learning partial differential equations

Reference 31

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Physics informed neural network using finite difference method

Reference 32

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Morphogenesis beyond in vivo

Reference 33

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning SGDR: Stochastic Gradient Descent with Warm Restarts

Reference 34

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

Reference 35

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This paper cites Deep learning for universal linear embeddings of nonlinear dynamics.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Deep learning for universal linear embeddings of nonlinear dynamics

Reference 36

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This paper cites A review of automatic differentiation and its efficient implementation.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning A review of automatic differentiation and its efficient implementation

Reference 37

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Observation caef5a3b-0423-4b90-9874-60c0d1b7f46c · outbound

This paper cites Fourier continuation for exact derivative computation in physics-informed neural operators.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Fourier continuation for exact derivative computation in physics-informed neural operators

Reference 38

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Discrete mollification and automatic numerical differentiation

Reference 39

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning On estimating regression

Reference 40

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This paper cites A review on the attention mechanism of deep learning.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning A review on the attention mechanism of deep learning

Reference 41

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Stochastic weather and climate models

Reference 42

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Automatic differentiation in pytorch

Reference 43

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Thermodynamically consistent physics-informed neural networks for hyperbolic systems

Reference 44

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This paper cites Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN

Reference 45

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

Reference 46

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Observation cdd72ca4-9ea4-4f89-ad33-9ac251d652fe · outbound

This paper cites Chromatin fibers are formed by heterogeneous groups of nucleo- somes in vivo.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Chromatin fibers are formed by heterogeneous groups of nucleo- somes in vivo

Reference 47

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This paper cites Neural ODE control for classification, approxima- tion, and transport.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Neural ODE control for classification, approxima- tion, and transport

Reference 48

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This paper cites Stochastic optical reconstruction mi- croscopy (STORM) provides sub-diffraction-limit image resolution.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Stochastic optical reconstruction mi- croscopy (STORM) provides sub-diffraction-limit image resolution

Reference 49

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.359203Z

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Observation 59be494d-df48-44ba-b4c1-63b5e2cce390 · outbound

This paper cites Smoothing and differentiation of data by simplified least squares procedures.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Smoothing and differentiation of data by simplified least squares procedures

Reference 50

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Observation 8401c7da-1113-4773-88a4-e9e419afc4e6 · outbound

This paper cites Super-resolution microscopy demystified.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Super-resolution microscopy demystified

Reference 51

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.335800Z

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Observation 74eb8d85-bfd2-412a-b043-31f2a98c79bd · outbound

This paper cites Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approxi- mation power.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approxi- mation power

Reference 52

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.324269Z

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Observation e80e13d3-eede-41fc-af8c-87a950b8f883 · outbound

This paper cites Inverse problem theory and methods for model parameter estimation.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Inverse problem theory and methods for model parameter estimation

Reference 53

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Observation f7198408-62a2-4b7f-ab1b-943b22c49a1a · outbound

This paper cites From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning

Reference 54

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This paper cites Attention is all you need.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Attention is all you need

Reference 55

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This paper cites Polymer model integrates imaging and sequencing to reveal how nanoscale heterochromatin domains influence gene expression.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Polymer model integrates imaging and sequencing to reveal how nanoscale heterochromatin domains influence gene expression

Reference 56

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.298601Z

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This paper cites Understanding and mitigating gradient flow pathologies in physics-informed neural networks.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Understanding and mitigating gradient flow pathologies in physics-informed neural networks

Reference 57

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.288455Z

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This paper cites On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural net- works.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural net- works

Reference 58

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.278296Z

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This paper cites PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks

Reference 59

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This paper cites Latent Neural Operator for Solving Forward and Inverse PDE Problems.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Latent Neural Operator for Solving Forward and Inverse PDE Problems

Reference 60

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Smooth regression analysis

Reference 61

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.268045Z

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This paper cites Computational models for mechanics of morphogenesis.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Computational models for mechanics of morphogenesis

Reference 62

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.256710Z

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This paper cites Understanding and improving layer normalization.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Understanding and improving layer normalization

Reference 63

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This paper cites Parameter Estimation of Partial Differential Equation Models.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Parameter Estimation of Partial Differential Equation Models

Reference 64

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This paper cites B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data

Reference 65

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.235153Z

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This paper cites Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems

Reference 66

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This paper cites Coupled data assimilation and parameter estimation in coupled ocean–atmosphere models: a review.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Coupled data assimilation and parameter estimation in coupled ocean–atmosphere models: a review

Reference 67

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This paper cites PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks

Reference 68

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This paper cites Elsevier, 2005.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Elsevier, 2005

Reference 69

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.203015Z

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This paper cites They can be incorporated into any layer by modifying the standard transformation in a multi-layer perceptron (MLP).

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning They can be incorporated into any layer by modifying the standard transformation in a multi-layer perceptron (MLP)

Reference 72

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.192555Z

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Unresolved cited work

Reference 73

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This paper cites Given an inputx, we define a set of frequenciesωi within an intervalI⊂ R.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Given an inputx, we define a set of frequenciesωi within an intervalI⊂ R

Reference 74

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.172138Z

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Reference 75

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This paper cites A common approach for this adjustment is cosine annealing, which schedules the learning rate to follow a cosine curve, ensuring a smooth and gradual decay.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning A common approach for this adjustment is cosine annealing, which schedules the learning rate to follow a cosine curve, ensuring a smooth and gradual decay

Reference 76

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.150375Z

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Reference 77

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Reference 78

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Observation 42f681cb-3598-47c7-8442-c4c8996f71bd · outbound

This paper cites For the Langevin equation with constant noisy parameter, various configurations accurately estimate the mean Λ (Fig.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning For the Langevin equation with constant noisy parameter, various configurations accurately estimate the mean Λ (Fig

Reference 79

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.117339Z

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Reference 80

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Unresolved cited work

Reference 81

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Unresolved cited work

Reference 82

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This paper cites After applying Watson Kernel interpolation, the interpolated values are used to update image features such asϕh,ϕe, andϕn.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning After applying Watson Kernel interpolation, the interpolated values are used to update image features such asϕh,ϕe, andϕn

Reference 83

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:57:35.071122Z

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Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning Unresolved cited work

Reference 191

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This paper cites URL: https://doi.org/10.1038/s42256- 021-00302-5.

Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning URL: https://doi.org/10.1038/s42256- 021-00302-5

Reference 5839

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