InPhyRe: Physics-Informed Reconstruction Methods

• InPhyRe is a polysemous concept representing physics-informed frameworks that embed explicit physical models into neural reconstruction, inverse rendering, and reasoning tasks.
• Applications include wireless imaging, PET reconstruction, intrinsic image decomposition, and acoustic phase retrieval, each leveraging physics constraints to guide latent inference.
• Practical insights show that combining learned representations with physics-based models enhances stability and data consistency, despite challenges in computational complexity.

InPhyRe is not a single standardized method or acronym in the recent literature. Instead, it denotes several technically distinct constructs that share a recurrent concern with embedding physics into inference, reconstruction, or reasoning. In one usage it is a physics-informed implicit neural representation for RIS-aided ISAC wireless imaging; in another it is Intrinsic Physics-based Reconstruction for intrinsic image decomposition; elsewhere it names a physically based inverse rendering framework for PET reconstruction, and in a different subfield it denotes a benchmark for inductive physical reasoning in large multimodal models. Other papers explicitly describe their methods as realizing an InPhyRe-style concept even when the acronym is not used in the title, notably in acoustic phase retrieval from magnitude-only data (Huang et al., 21 Jan 2026, Baslamisli et al., 2020, Li et al., 27 Aug 2025, Sreekumar et al., 12 Sep 2025, Schrader et al., 27 Jan 2026).

1. Terminology and conceptual scope

The term is therefore polysemous. A precise reading depends on domain, forward model, and task formulation. In reconstruction-oriented work, InPhyRe usually denotes a framework in which unknown latent variables are optimized under a differentiable physical model, a PDE constraint, or both. In the multimodal-reasoning literature, by contrast, InPhyRe refers to an evaluation benchmark rather than a solver (Huang et al., 21 Jan 2026, Sreekumar et al., 12 Sep 2025).

Usage of “InPhyRe”	Meaning in the source	Representative paper
Wireless imaging	Physics-Informed Implicit Neural Representation	(Huang et al., 21 Jan 2026)
Intrinsic decomposition	Intrinsic Physics-based Reconstruction	(Baslamisli et al., 2020)
PET reconstruction	physically based inverse rendering framework for PET image reconstruction	(Li et al., 27 Aug 2025)
Physical reasoning benchmark	Inductive Physical Reasoning	(Sreekumar et al., 12 Sep 2025)
Acoustic field recovery	physics-informed phase retrieval concept realized by PRB-PINN	(Schrader et al., 27 Jan 2026)

This heterogeneity suggests a family resemblance rather than a canonical architecture. The dominant family resemblance is the use of explicit physics to restrict an otherwise ill-posed inverse problem.

2. Shared inverse-problem structure

Across the reconstruction-oriented usages, the unknown is typically represented as a continuous field, image, or volumetric parameter map, and the learning objective combines measurement consistency with physically grounded constraints. The exact mechanism varies. In acoustic magnitude-field reconstruction, the complex pressure is written as $$u(\mathbf{x}) = A(\mathbf{x}) e^{i\phi(\mathbf{x})}$$, and the reconstructed field is constrained by the homogeneous Helmholtz equation,

$\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$

while data fidelity is imposed only on sparse magnitude samples in the dB domain. The total objective is

$$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$

with no boundary-condition term in that work (Schrader et al., 27 Jan 2026).

Wireless InPhyRe uses a different mechanism. The unknown ROI image is a scattering-coefficient field $\sigma$ represented by an INR $\mathcal{M}_{\boldsymbol\theta}$, and physics is enforced through a differentiable multipath forward model $f(\boldsymbol{\sigma},\boldsymbol{\Omega})$ rather than a PDE residual. The optimization problem is posed as NN parameter estimation under data fidelity and sparsity,

$$L_{\boldsymbol{\theta}}=\|\widehat{\mathbf{y}}_{\boldsymbol{\theta}}-\mathbf{y}\|_2+\alpha\|\widehat{\boldsymbol{\sigma}}_{\boldsymbol{\theta}}\|_1,$$

with $$\widehat{\mathbf{y}}_{\boldsymbol{\theta}}=f(\widehat{\boldsymbol{\sigma}}_{\boldsymbol{\theta}},\boldsymbol{\Omega})$$ (Huang et al., 21 Jan 2026).

In PET, the same pattern appears in inverse-rendering form. The unknown voxel activity image $I$ is mapped to expected sinogram counts by a differentiable forward operator $R(I)$, and the measurements are modeled as Poisson variables,

$\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$0

with gradients obtained by reverse-mode autodiff through Monte Carlo sub-LOR sampling, PSF perturbation, normalization, attenuation, and optional TOF weighting (Li et al., 27 Aug 2025).

In accelerated MRF reconstruction, the unknown time-series magnetization image $\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$1 is constrained simultaneously by the acquisition model and by the Bloch response manifold. The stated optimization problem is

$\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$2

where $\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$3, $\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$4, and $\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$5 is supplied by a pretrained denoising diffusion prior (Mayo et al., 29 Jun 2025).

These formulations show that physics-informedness in the InPhyRe literature is not synonymous with PINN residual minimization. It can mean PDE enforcement, differentiable rendering, forward-model supervision, dictionary projection, or coupled multi-physics constraints.

3. Wave and field reconstruction

The most explicit reconstruction use of the InPhyRe concept appears in wave-field problems. The acoustic phase-retrieval PINN of 2026 reconstructs the spatial magnitude of an acoustic pressure field from sparse magnitude-only data by jointly learning two neural fields, one for magnitude and one for phase, and enforcing the Helmholtz equation on the reconstructed complex field. The study uses a $\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$6 room, a $\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$7 target cube discretized as a $\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$8 lattice, 64 random point sources outside the region of interest, single-frequency runs at 200, 400, and 600 Hz, and random sensor subsets with $\nabla^2 u(\mathbf{x}) + k^2 u(\mathbf{x}) = 0,$9. Each of the two MLPs has 4 hidden layers with 256 neurons per layer and tanh activations; both consume random Fourier features with $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$0, and training uses AdamW for $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$1 iterations with $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$2 and $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$3. Across all tested frequencies and sensor counts, the proposed PRB-PINN achieves the lowest test data loss, outperforming both nearest-neighbor interpolation and a neural field without physics (Schrader et al., 27 Jan 2026).

In RIS-aided ISAC wireless imaging, InPhyRe is explicitly the method name. The INR maps continuous 3D coordinates to scattering coefficients, uses a six-layer MLP with 256 neurons per layer, sine activations, a Sigmoid output layer, and Fourier-feature positional encoding with $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$4. Training proceeds by predicting scattering coefficients on a training grid, passing them through a differentiable multipath forward model, and minimizing the combined data-consistency and sparsity objective. The method is resolution-agnostic because the learned INR can be queried on any output grid after training. Under the consistent comparison setting described in the paper—single-antenna TX/RX, TX–ROI–RIS–RX path only, DFT RIS phases, and $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$5—InPhyRe with $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$6 achieves high-quality images and outperforms FT and CS even with $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$7. The ablations also report that “sin + positional encoding” achieves SSIM $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$8 in 5000 epochs and near-perfect images within about 200 epochs (Huang et al., 21 Jan 2026).

Related work broadens this wave-physics lineage. PE-PINN is presented as a core solver for an InPhyRe-style large-scale wave reconstruction pipeline: it embeds plane-wave and spherical-wave carrier kernels directly into the architecture, separates incident and scattered fields, and uses material-aware domain decomposition. In a 2D $$\mathcal{L}_{\text{total}}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}},$$9, 2.4 GHz free-space example, PE-PINN converges in 50,000 iterations and 17 m 54 s with $\sigma$0, whereas a vanilla PINN required 10,000,000 iterations and 26 h 27 m and did not converge on the imaginary part; the reported 3D room-scale cases also imply several orders of magnitude lower memory use than FEM (Zhang et al., 13 Feb 2026). APINN extends the same physics-informed reconstruction logic to radiative transfer by introducing auxiliary outputs $\sigma$1 that encode Legendre projections of the intensity field, thereby replacing angular quadrature with differential constraints. In slab RTE benchmarks, APINN reaches final losses around $\sigma$2 in isotropic and Rayleigh cases and retrieves single-scattering albedo from only two boundary mean-intensity measurements with relative errors down to $\sigma$3 (Riganti et al., 2023).

4. Imaging, tomography, and inverse rendering

In intrinsic image decomposition, InPhyRe stands for Intrinsic Physics-based Reconstruction. The method begins with photometric invariants derived from a diffuse Lambertian model. The albedo gradient index,

$\sigma$4

masks albedo transitions, while the shading gradient index

$\sigma$5

is used only on homogeneous patches. A sparse shading estimate is recovered by least-squares integration and then densified via the convex quadratic objective

$\sigma$6

A dual-encoder, dual-decoder CNN then refines shading and predicts albedo under image-formation consistency. Reported results include MIT Intrinsics shading MSE 0.0069 and LMSE 0.0418, NIR-RGB shading MSE 0.0017, MIII shading MSE 0.0002, and IIW WHDR 26.8% with guided-filter post-processing (Baslamisli et al., 2020).

PET InPhyRe reformulates reconstruction as inverse rendering. Implemented in Dr.Jit, it combines Monte Carlo sub-LOR sampling inside detector crystals with an analytical projector, Gaussian PSF perturbation, normalization, attenuation correction, and autodiff-based voxel gradients. In phantom studies using MLEM, InPhyRe yields higher SNR than CASToR in the Derenzo phantom, lower gray-matter COV and lower ventricle spill-over ratio than CASToR in the Hoffman phantom, and on human $\sigma$7MK6240 brain PET produces hippocampal SUVR $\sigma$8 versus $\sigma$9 for the Siemens reconstruction, together with a 4.5× speed-up on the Hoffman phantom and 9.75× on the hotspot phantom (Li et al., 27 Aug 2025).

The InPhyRe logic also appears in quantitative MRI. MRF-DiPh enforces both k-space consistency and Bloch-response consistency during diffusion-based reconstruction. It uses a conditional diffusion UNet within an HQS/ADMM loop, a dictionary with 95k atoms simulated by EPG, subspace compression to $\mathcal{M}_{\boldsymbol\theta}$0, and one ADMM iteration per diffusion step. On accelerated brain MRF with $\mathcal{M}_{\boldsymbol\theta}$1, the base model reports MAPE 6.75 for T1 and 18.40 for T2, TSMI NRMSE 18.65, and k-space NRMSE 22.82, improving over MRF-IDDPM and other baselines (Mayo et al., 29 Jun 2025). PHIMO applies a similar physics-informed principle to motion correction in multi-echo GRE by learning slice-wise line-exclusion masks whose retained reconstructions best satisfy mono-exponential $\mathcal{M}_{\boldsymbol\theta}$2 decay; the method approaches the performance of the redundant-acquisition state-of-the-art while reducing acquisition time by over 40% (Eichhorn et al., 2024).

Single-hologram X-ray phase retrieval supplies a self-supervised generative variant. The method called SelfPhish in the paper, treated there as an InPhyRe-style approach, reconstructs both absorbance $\mathcal{M}_{\boldsymbol\theta}$3 and phase $\mathcal{M}_{\boldsymbol\theta}$4 from a single propagation-based hologram by generating the exit wave $\mathcal{M}_{\boldsymbol\theta}$5, propagating it with a differentiable Fresnel operator, and matching the measured detector intensity. No paired, unpaired, or simulated training data are required. On simulated data, the phase reconstruction reaches MSSIM $\mathcal{M}_{\boldsymbol\theta}$6 at Fresnel number $\mathcal{M}_{\boldsymbol\theta}$7, and 512×512 holograms require about 1.5 minutes for 2000 iterations on an NVIDIA A100 GPU (Yang et al., 21 Aug 2025).

5. Physical plausibility, scene reconstruction, and reasoning

A broader extension of the same design philosophy appears in physically plausible neural scene reconstruction. PhyRecon jointly optimizes a NeuS-style SDF, differentiable rendering losses, and a differentiable particle-based simulator connected through Surface Points Marching Cubes. Physical stability is measured through contact-point motion under simulation, while rendering and physical uncertainty are used to weight losses and guide pixel sampling toward slender, support-critical structures. Across ScanNet++, ScanNet, and Replica, the method improves both geometric metrics and physical stability; object-level stability ratio rises from 25.28% to 78.16% on ScanNet++, from 26.56% to 70.31% on ScanNet, and from 30.26% to 77.63% on Replica (Ni et al., 2024).

The benchmark paper “InPhyRe Discovers” uses the acronym differently. Here InPhyRe denotes a visual question answering benchmark for inductive physical reasoning in large multimodal models. It contains ten synthetic collision scenarios—three regular and seven irregular—with about 2,000 samples per scenario, for a total of about 20,000 samples. Evaluation is four-way multiple choice, accuracy is the sole metric, and few-shot prompting can use either video+text exemplars or video-only exemplars. The main finding is negative: among 39 zero-shot evaluations on regular scenarios, accuracy exceeds 80% in only 6 cases, and switching from video+text to video-only exemplars often collapses performance, indicating strong language bias and weak use of visual demonstrations (Sreekumar et al., 12 Sep 2025).

The juxtaposition is instructive. In most of the cited literature, InPhyRe names a reconstruction framework that uses physics to reduce ambiguity in latent-variable inference. In the multimodal benchmark, it instead names an evaluation protocol designed to test whether a model can infer and adapt to new physical rules from demonstrations. The two senses share a concern with physics-guided generalization, but they are methodologically distinct.

6. Limitations, ambiguities, and future directions

A recurring limitation is that physics-informedness does not eliminate non-identifiability. The acoustic phase-retrieval paper explicitly notes that phase is not uniquely identifiable from magnitude-only data, so the predicted phase need not match ground truth even when the reconstructed magnitude improves (Schrader et al., 27 Jan 2026). The wireless imaging formulation treats unknown higher-order multipaths as disturbance and assumes quasi-static channels and a static environment, so severe model mismatch can degrade reconstruction (Huang et al., 21 Jan 2026). PET InPhyRe currently addresses randoms and scatter through preprocessing rather than explicit forward modeling, and the authors identify computational cost, Monte Carlo gradient variance, and future modeling of Compton scatter, positron range blurring, and photon acollinearity as open issues (Li et al., 27 Aug 2025).

MRI and holographic variants expose a different limitation: physics-consistency steps can dominate runtime. In MRF-DiPh, the data-consistency proximal and dictionary matching account for most of the 44 s reconstruction time in the base setting, and the authors note that scaling to 3D may require approximate matching or faster solvers (Mayo et al., 29 Jun 2025). In single-hologram X-ray phase retrieval, the method depends on the Fresnel-regime assumptions, appropriate padding to satisfy sampling constraints, and early stopping to avoid overfitting noise; flat-field inaccuracies can manifest as low-frequency phase background (Yang et al., 21 Aug 2025).

The term itself is also unstable. One misconception would be to treat InPhyRe as a single named architecture. The record instead shows several incompatible expansions and task definitions. Another misconception would be to infer robust physical reasoning from few-shot gains alone. The InPhyRe benchmark for LMMs shows that few-shot improvements can derive largely from textual exemplar bias rather than visual induction, especially when video+text exemplars are replaced by video-only exemplars (Sreekumar et al., 12 Sep 2025).

Future directions in the cited papers are correspondingly diverse: multi-frequency joint training and boundary-condition residuals for acoustic magnitude reconstruction, multi-RIS and multi-frequency operation for wireless imaging, dynamic PET and richer transport physics for inverse rendering, and broader quantitative MRI extensions via alternative response manifolds or more faithful sequence models (Schrader et al., 27 Jan 2026, Huang et al., 21 Jan 2026, Li et al., 27 Aug 2025, Mayo et al., 29 Jun 2025). This suggests that InPhyRe is best understood not as a settled framework but as a moving label for methods that attempt to couple learned representations with explicit physical structure, sometimes for reconstruction, sometimes for inverse rendering, and sometimes for evaluating whether models can reason beyond the physics encoded in their parameters.