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arxiv: 2606.21945 · v1 · pith:X3NNQ7AQnew · submitted 2026-06-20 · ⚛️ physics.optics

Beyond Data-Driven: How Physics-Informed Neural Networks are Reshaping Multi-Physics Design and Discovery

Amir H. M. Labeb , Basmala Sallam , Abdelrahman W. Elsayed , Islam I. Abdulaal , Amany M. Kamal , Omar A. M. Abdelraouf This is my paper

Pith reviewed 2026-06-26 11:53 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords physics-informed neural networksPINNsmulti-physics modelingnanophotonicsequation discoveryscientific machine learninginverse designdigital twins
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The pith

Embedding governing equations as soft constraints in neural network training confines solutions to the physics manifold and enables modeling with sparse data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews physics-informed neural networks that incorporate partial differential equations and conservation laws directly into the loss function during training. This reformulates learning as a constrained optimization task rather than pure data fitting. The approach supports forward problems, inverse design, and equation discovery in fields such as nanophotonics and fluid mechanics where measurements are limited. It examines integration of physical knowledge across data generation, architecture choices, and loss design while comparing to traditional numerical methods.

Core claim

Physics-informed neural networks reformulate the learning task as a constrained optimization problem in which admissible solutions are confined to the manifold defined by the underlying physics, enabling the construction of models that are simultaneously data-efficient, physically consistent, and capable of operating in regimes where measurements are sparse or indirect.

What carries the argument

Embedding partial differential equations, conservation laws, and constitutive relationships as soft constraints in the training objective.

Load-bearing premise

Embedding governing equations as soft constraints in the training objective confines solutions to the physics-defined manifold without introducing significant accuracy or convergence trade-offs.

What would settle it

Training a PINN on a problem with known exact solution and observing that the network output violates a conservation law by more than numerical tolerance would falsify the claim of confinement to the physics manifold.

Figures

Figures reproduced from arXiv: 2606.21945 by Abdelrahman W. Elsayed, Amany M. Kamal, Amir H. M. Labeb, Basmala Sallam, Islam I. Abdulaal, Omar A. M. Abdelraouf.

Figure 1
Figure 1. Figure 1: Overview of PINNs: applications and paradigms. 1.3 Challenges and opportunities in integrating different physical disciplines While the formal PINNs framework is largely agnostic to the specific form of the governing equations, its practical deployment reveals domain-specific challenges arising from the analytical and numerical properties of different PDE systems. In fluid mechanics, the incompressible and… view at source ↗
Figure 2
Figure 2. Figure 2: PINNs framework for forward electromagnetic modeling and representative results. (a) Illustration of the physics-driven loss formulation, where the residuals of Maxwell’s equations are evaluated under two electric fields: the electric field obtained using the first-order Born approximation and the ground-truth solution computed using COMSOL [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Application of ML for solving NSE: a) average velocity contour for a turbulent square duct flow at Re = 3200. b) Data comparison for Surface pressure coefficient for NACA 0012 airfoil. c) Comparison of velocity contour for a lid￾driven cavity at Re = 400. d) Turbulent kinetic energy contour for k-ω-PINNs compared with DNS and k-ω turbulence model [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Some of the Forward Design PINNs architectures in Various specializations, and comparing the PINN model with other models. (a) PINN architectures In Cosmology.[143] (b) PINN architectures in Space Weather.[147] (c) PINN architectures in Astronomy.[138] (d) Kurucz-a1 neural network architecture with dual-encoder design.[153] (e) Comparing the total evaluation time of PINN with other gravity models.[151] (f)… view at source ↗
Figure 5
Figure 5. Figure 5: PINN approaches for arterial hemodynamics modelling. a) Topology of arterial network with 3 vessels shown and 1 bifurcation point.[163] b) The implementation of the parabolic inlet velocity profile and evaluated PINN architectures on four distinct geometries. These included two idealized models (a cylinder and a bifurcation) and two patient-specific intracranial aneurysm models.[164] Beyond the modelling o… view at source ↗
Figure 6
Figure 6. Figure 6: Problem setup of Ref [168] for cancer detection using thermal analysis. a) 2D cylindrical computational domain used for thermal analysis. b) Example of the random sampling of interior collocation points (blue) and boundary points (orange) used to train the PINN solver for the bioheat equation. In addition to cancer treatments and predictions, PINNs were also used in thermal forward modelling, which is used… view at source ↗
Figure 7
Figure 7. Figure 7: PINNs frameworks for inverse electromagnetic modeling and parameter retrieval. (a) Schematic illustration of a PINN used to solve the parameter retrieval problem in near-field microscopy. A fully connected neural network (FCNN) takes spatial coordinates and trainable parameters as inputs and outputs the surrogate solution of the governing partial differential equation (PDE). The loss function enforces cons… view at source ↗
Figure 8
Figure 8. Figure 8: Hidden Boundary Detection a) Comparing PINN-derived h, u, v, and p to reference simulation findings using the spectral element approach for forced convection in enclosures. b) Quantitative hemodynamic prediction in a three￾dimensional intracranial aneurysm. Flow field in microfluidic devices is frequently studied using Particle Image Velocimetry (PIV) or Lagrangian Particle Tracking. Shin et al. (2025) [18… view at source ↗
Figure 10
Figure 10. Figure 10: Comparing results between PINN and reference simulation at different angle of attack. 4.2.2 Viscosity and permeability prediction Inverse design involves detecting material and transport parameters such as fluid viscosity or porous media permeability via indirect observations. Li et al. (2022) [188]investigate the reconstruction of thermal fluid fields for nanofluids using physics-informed Deep learning. … view at source ↗
Figure 11
Figure 11. Figure 11: Characterization of porous media using PINNs. (a) Diagrammatic workflow showing how experimental data and the governing diffusion equation are coupled within a deep learning framework to forecast a non-uniform FVF distribution. (b) Verification of PINN predictions for permeability and compaction stress using experimental data and traditional empirical models (power law and modified Carman-Kozeny) [PITH_F… view at source ↗
Figure 12
Figure 12. Figure 12: Some of the Inverse Design PINNs architectures in Various specializations, and comparing the PINN model with other models. (a) PINN architectures In Astronomy.[192] (b) PINN architectures in Cosmology.[195] (c) PINN architectures in Space Weather.[200] (d) Newtonian model of the chirping waveform emitted by a system of two merging equal masses m = 30 M⊙. Both polarizations are shown.[205] (e) Evolved magn… view at source ↗
Figure 13
Figure 13. Figure 13: Clinical applications and multiparametric map reconstructions utilizing PINNs. a) Joint estimation of B0 -field maps and structural images in low-field MRI[224]. b) Fast, high resolution quantitative mapping of Fat Fraction and water￾T2 in lower limb muscles for the assessment of neuromuscular disorders using Extended Phase Graph (EPG) theory within [PITH_FULL_IMAGE:figures/full_fig_p052_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of PINN, PDE-Net2, and SINDy for reconstructing ionospheric Total Electron Content (TEC) dynamics from Step 1 to Step 2. The results show that PDE-Net2 achieves the closest agreement with the true TEC distribution, outperforming PINN and SINDy in capturing ionospheric structures.[235] 5.4 Data-Driven Equation Discovery in Biomedical Engineering In the biomedical field, some effort has been dedi… view at source ↗
read the original abstract

Physics-informed neural networks (PINNs) constitute a rapidly maturing class of scientific machine learning models in which the governing equations of a physical system are embedded directly into the training objective as soft constraints. By enforcing partial differential equations (PDEs), conservation laws, and constitutive relationships during optimization, PINNs enable the construction of models that are simultaneously data-efficient, physically consistent, and capable of operating in regimes where measurements are sparse or indirect. In contrast to conventional deep learning, where the loss is typically defined solely in terms of data misfit, the learning task in PINNs is reformulated as a constrained optimization problem in which admissible solutions are confined to the manifold defined by the underlying physics. This review provides a comprehensive assessment of recent developments in physics-informed machine learning with an emphasis on PINN-based formulations for forward modelling, inverse design, and equation discovery across nanophotonics, fluid mechanics, astronomy, and biomedical engineering. Particular attention is devoted to how physical knowledge is injected at different stages of the modelling pipeline, including synthetic data generation, non-dimensionalization and scaling, architecture selection, loss design, and post-training regularization. We highlight emerging strategies for multi-physics coupling, transfer learning across parameter and geometry spaces, and rigorous benchmarking against established numerical solvers. Finally, the review discusses interpretability, uncertainty quantification, and hardware acceleration, and articulates how physics-informed learning is reshaping engineering practice by enabling digital twins and design workflows that combine simulation and data in a unified differentiable framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript is a review of physics-informed neural networks (PINNs) that claims embedding PDEs, conservation laws, and constitutive relations as soft constraints in the training loss reformulates the problem as constrained optimization, confining solutions to the physics-defined manifold. This is said to yield data-efficient, physically consistent models suitable for sparse-data regimes in forward modeling, inverse design, and equation discovery across nanophotonics, fluid mechanics, astronomy, and biomedical engineering. The review surveys injection of physics at stages including data generation, architecture choice, loss design, multi-physics coupling, transfer learning, benchmarking, interpretability, and hardware acceleration.

Significance. A balanced review that rigorously benchmarks PINN formulations against established solvers and discusses multi-physics coupling strategies could be useful for practitioners. The abstract's emphasis on non-dimensionalization, scaling, and post-training regularization identifies practical implementation points that are often under-discussed.

major comments (1)
  1. [Abstract] Abstract: The central claim that soft-constraint embedding 'confines admissible solutions to the manifold defined by the underlying physics' is load-bearing for the review's thesis on data-efficiency and consistency with sparse measurements. The weighted sum of data-misfit and residual losses does not guarantee manifold membership; in multi-physics settings with disparate length scales or stiffness, this formulation frequently produces solutions satisfying neither term to high accuracy. The review must explicitly address convergence trade-offs and contrast with hard-constraint or augmented-Lagrangian alternatives.
minor comments (2)
  1. [Abstract] The abstract paragraph on reformulation as constrained optimization is repeated in substance later in the text; consolidate to avoid redundancy.
  2. When citing external benchmarks, ensure each comparison specifies the exact PDE residual norm and data density used, rather than qualitative statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback on our review manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that soft-constraint embedding 'confines admissible solutions to the manifold defined by the underlying physics' is load-bearing for the review's thesis on data-efficiency and consistency with sparse measurements. The weighted sum of data-misfit and residual losses does not guarantee manifold membership; in multi-physics settings with disparate length scales or stiffness, this formulation frequently produces solutions satisfying neither term to high accuracy. The review must explicitly address convergence trade-offs and contrast with hard-constraint or augmented-Lagrangian alternatives.

    Authors: We agree that the abstract phrasing presents the soft-constraint formulation in terms that could be read as implying strict manifold confinement, whereas the weighted loss provides only an approximation whose quality depends on weighting, optimization, and problem conditioning. This point is valid and the manuscript does not currently contain an explicit discussion of these trade-offs in the abstract or a dedicated contrast with hard-constraint or augmented-Lagrangian formulations. In the revised manuscript we will (i) rephrase the abstract to state that admissible solutions are encouraged to lie near the physics manifold and (ii) insert a new subsection that reviews known convergence limitations of soft-constraint PINNs in multi-physics problems with disparate scales or stiffness and that briefly contrasts these with hard-constraint and augmented-Lagrangian alternatives. These changes will be made without altering the overall scope or conclusions of the review. revision: yes

Circularity Check

0 steps flagged

No circularity: review surveys external literature without internal derivations or self-referential claims

full rationale

This is a survey paper assessing PINN developments in multiple fields. The abstract and provided text describe existing techniques (embedding PDEs as soft constraints, reformulating learning as constrained optimization) by reference to prior work, without presenting new equations, fitted parameters, predictions, or derivations that could reduce to the paper's own inputs. No self-citations are invoked as load-bearing uniqueness theorems, and no ansatzes or renamings are introduced. The central claims rest on external benchmarking and literature, making the text self-contained against external sources with no reduction by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The review rests on standard domain assumptions about PINN formulations drawn from the surveyed literature; no free parameters or new entities are introduced by the review itself.

axioms (1)
  • domain assumption Embedding governing equations as soft constraints in the loss confines admissible solutions to the physics manifold
    Invoked in the abstract when describing the reformulation of the learning task as constrained optimization.

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Forward citations

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