Paper Citation Record · LEDGER

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations

As of 7 August 2026, this Paper Citation Record lists 35 of 35 outbound references and 0 inbound Pith citation observations for arXiv:2607.19377.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2607.19377 v1

Coverage vector

measured 35 of 35 reference resolution

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Source: cited_works

Reference resolution

35 of 35 outbound references displayed

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Outbound references

Observation d3484cbb-f80b-4266-b312-593ebf903250 · outbound

This paper cites Evans.Partial Differential Equations.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Evans.Partial Differential Equations

Reference 1

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Observation 53a9b028-8f40-44ae-9939-c07e9c674bee · outbound

This paper cites LeVeque.Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations LeVeque.Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems

Reference 2

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Observation 37b54676-c56e-4b5f-8fc7-53a66685fa03 · outbound

This paper cites Inverse problem solution and optimization in the vibration analysis of nanocomposite cylindrical shell using l-bfgs-b algorithm.Composite Structures, 370:119309, 2025.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Inverse problem solution and optimization in the vibration analysis of nanocomposite cylindrical shell using l-bfgs-b algorithm.Composite Structures, 370:119309, 2025

Reference 3

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Observation 5421799d-85a4-41f7-b048-4824c7f03e44 · outbound

This paper cites an unresolved cited work.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Unresolved cited work

Reference 4

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Observation 5aa6677c-efea-4c34-8e7a-70fe366b1aa9 · outbound

This paper cites an unresolved cited work.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Unresolved cited work

Reference 5

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Observation 6125c93d-9e62-4a06-8e64-8d10c1ef9d6a · outbound

This paper cites Springer, 2016.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Springer, 2016

Reference 6

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Observation 534f68b6-b32a-43e5-9c40-b292e73ee9de · outbound

This paper cites Why starting from differential equations for computational physics?Journal of Com- putational Physics, 257:1260–1290, 2014.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Why starting from differential equations for computational physics?Journal of Com- putational Physics, 257:1260–1290, 2014

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.938593Z

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Observation 94a4b8f9-ed9b-4f6d-9d7e-2c2eef1651d4 · outbound

This paper cites An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology.Journal of Computational Physics, 133(2):289–309, 1997.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology.Journal of Computational Physics, 133(2):289–309, 1997

Reference 8

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Observation 22d28b7f-ec07-4a54-b622-d937ce000243 · outbound

This paper cites Finite-difference time-domain methods.Nature Reviews Methods Primers, 3(1):75, 2023.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Finite-difference time-domain methods.Nature Reviews Methods Primers, 3(1):75, 2023

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.944861Z

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Observation 79bbce95-943f-4f6a-959e-647fca103c16 · outbound

This paper cites The finite volume method.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations The finite volume method

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.947537Z

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Observation 377e30bf-93b4-470d-a6f2-ab0e5af1ae10 · outbound

This paper cites John Wiley & Sons, 2012.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations John Wiley & Sons, 2012

Reference 11

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Observation 20f32cb4-771b-465f-99b2-8310600ca919 · outbound

This paper cites SIAM, 2006.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations SIAM, 2006

Reference 12

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Observation 9d78b921-53f5-43f6-af03-5cbd202743fc · outbound

This paper cites Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

Reference 13

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Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Unresolved cited work

Reference 14

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Observation 888ed3c4-7b1a-411f-9739-d688edba0671 · outbound

This paper cites Utilizing optimal physics-informed neural networks for dynamical analysis of nanocomposite one-variable edge plates.Thin-Walled Structures, 202:111928, 2024.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Utilizing optimal physics-informed neural networks for dynamical analysis of nanocomposite one-variable edge plates.Thin-Walled Structures, 202:111928, 2024

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.961692Z

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Observation 3491b359-a42e-465b-a88b-ffcc900fa986 · outbound

This paper cites St-pinn: shared-trunk pinns for discontinuous multi-domain pdes.Engineering with Computers, 42(4):113, 2026.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations St-pinn: shared-trunk pinns for discontinuous multi-domain pdes.Engineering with Computers, 42(4):113, 2026

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.964278Z

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Observation 2555b102-e5b4-44e9-95d4-6a270d6585e5 · outbound

This paper cites Scientific machine learning through physics–informed neural networks: Where we are and what’s next.Journal of Scientific Computing, 92(3):88, 2022.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Scientific machine learning through physics–informed neural networks: Where we are and what’s next.Journal of Scientific Computing, 92(3):88, 2022

Reference 17

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.966750Z

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Observation 53142764-8c8d-44f8-8801-2d0a9627259b · outbound

This paper cites Physics-informed neural networks for pde problems: A comprehensive review.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Physics-informed neural networks for pde problems: A comprehensive review

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.969119Z

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Observation 87a1f56b-ad55-4b0a-9de3-c094850fe104 · outbound

This paper cites Understanding and mitigating gradient flow patholo- gies in physics-informed neural networks.SIAM Journal on Scientific Computing, 43(5):A3055– A3081, 2021.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Understanding and mitigating gradient flow patholo- gies in physics-informed neural networks.SIAM Journal on Scientific Computing, 43(5):A3055– A3081, 2021

Reference 19

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Observation a37cf7bf-514b-472f-9934-2d3ba41ef523 · outbound

This paper cites Charac- terizing possible failure modes in physics-informed neural networks.Advances in neural information processing systems, 34:26548–26560, 2021.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Charac- terizing possible failure modes in physics-informed neural networks.Advances in neural information processing systems, 34:26548–26560, 2021

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.974485Z

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Observation e8dee1dc-5c4b-4fe0-a7d8-8637bb8146d5 · outbound

This paper cites When and why pinns fail to train: A neural tangent kernel perspective.Journal of Computational Physics, 449:110768, 2022.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations When and why pinns fail to train: A neural tangent kernel perspective.Journal of Computational Physics, 449:110768, 2022

Reference 21

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.977110Z

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Observation 373a370c-baea-4327-991e-6c3cf5563b08 · outbound

This paper cites Artificial neural networks for solving ordinary and partial differential equations.IEEE transactions on neural networks, 9(5):987–1000, 1998.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Artificial neural networks for solving ordinary and partial differential equations.IEEE transactions on neural networks, 9(5):987–1000, 1998

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.979508Z

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Observation a120897d-4f5c-449c-84b7-88a8bf7f6a47 · outbound

This paper cites Self-adaptive physics-informed neural networks using a soft attention mechanism, 2021.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Self-adaptive physics-informed neural networks using a soft attention mechanism, 2021

Reference 23

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Observation e9b4ba92-90a0-4553-ab77-b034f985a7f5 · outbound

This paper cites Deepxde: A deep learning library for solving differential equations.SIAM review, 63(1):208–228, 2021.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Deepxde: A deep learning library for solving differential equations.SIAM review, 63(1):208–228, 2021

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:43.984498Z

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Observation e80d19e0-db99-477f-8256-715496d1dbda · outbound

This paper cites Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks.Computer Methods in Applied Mechanics and Engineering, 389:114333, 2022.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks.Computer Methods in Applied Mechanics and Engineering, 389:114333, 2022

Reference 25

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Observation 8ed15c4b-48af-44ec-96b5-bb6428eb8c0a · outbound

This paper cites Physics-informed neural networks with hard and soft boundary conditions for linear free surface waves.Physics of fluids, 37(8), 2025.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Physics-informed neural networks with hard and soft boundary conditions for linear free surface waves.Physics of fluids, 37(8), 2025

Reference 26

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Observation ffadcd08-9908-438b-956a-0b238fce24d1 · outbound

This paper cites Physics-Informed Neural Networks with Complementary Soft and Hard Constraints for Solving Complex Boundary Navier-Stokes Equations.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Physics-Informed Neural Networks with Complementary Soft and Hard Constraints for Solving Complex Boundary Navier-Stokes Equations

Reference 27

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Observation cffe228e-a44b-4c94-93ae-cd567bb7ed2a · outbound

This paper cites Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving

Reference 28

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Observation b49a6594-acf6-427a-8c62-f7915cf65049 · outbound

This paper cites Physics-informed neural networks with hard constraints for inverse design.SIAM Journal on Scientific Computing, 43(6):B1105–B1132, 2021.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Physics-informed neural networks with hard constraints for inverse design.SIAM Journal on Scientific Computing, 43(6):B1105–B1132, 2021

Reference 29

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This paper cites A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks.Journal of Computational Physics, 435: 110242, 2021.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks.Journal of Computational Physics, 435: 110242, 2021

Reference 30

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Observation de807b63-b4b8-4771-8624-cba377ce043e · outbound

This paper cites an unresolved cited work.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Unresolved cited work

Reference 31

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Observation cbc4a4a6-fdcb-4bce-a6b6-e50bfe4eedaa · outbound

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Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Automatic differentiation in pytorch

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:44.007709Z

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Observation 62ed0c28-8517-4e1c-8616-df6d28e57694 · outbound

This paper cites Pytorch: An imperative style, high- performance deep learning library.Advances in neural information processing systems, 32, 2019.

Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Pytorch: An imperative style, high- performance deep learning library.Advances in neural information processing systems, 32, 2019

Reference 33

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Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Adam: A Method for Stochastic Optimization

Reference 34

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Observation cdfcea3e-aeec-418c-924a-e4478b7bc5d7 · outbound

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Reliability-Aware Hard--Soft Physics-Informed Neural Networks for Robust Learning of Challenging Partial Differential Equations Improved adam optimizer for deep neural networks

Reference 35

Resolution: unresolved; no resolver link, observed 2026-08-02T09:16:44.016056Z

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