Physics-Informed Denoising Framework

• Physics-informed denoising frameworks are learning systems that integrate explicit physical constraints, such as conservation laws and PDE residuals, to ensure signal fidelity.
• They incorporate physical priors via loss regularization, process-level guidance, and architectural conditioning to adapt to domain-specific structures.
• Empirical results across microscopy, MRI, and sensing applications show improved metrics like PSNR and SNR, reduced artifacts, and robust performance in both zero-shot and paired setups.

A physics-informed denoising framework denotes a class of learning systems in which denoising is constrained by an explicit physical structure rather than learned solely from empirical input–output correlations. Across recent work, the relevant structure may be a governing PDE, an imaging forward operator, a conservation law, an algebraic sensor relation, a device model, or a feasibility region. In these frameworks, physics is not merely used for post hoc validation; it is inserted into the training loss, the reverse diffusion dynamics, the network conditioning pathway, the self-supervision strategy, or the construction of synthetic training data. This design pattern appears in microscopy, atomic-resolution STEM, smart-grid data synthesis, ultrasound compounding, multiparametric MRI reconstruction, and real-life sensing systems (Li et al., 2023, Awan et al., 2024, Wang et al., 24 Apr 2025, Asgariandehkordi et al., 26 Jun 2025, Mayo et al., 29 Jun 2025, Zhang et al., 2023).

1. Definition and conceptual scope

Physics-informed denoising is not a single model family. It is an umbrella formulation in which a denoiser is asked to satisfy two objectives simultaneously: statistical consistency with observed data and adherence to a physical prior. In the surveyed literature, the prior may encode AC power-flow feasibility, microscope point-spread formation, k-space measurement consistency, Bloch-response constraints, transport-of-intensity reconstruction physics, heat-conduction surrogates, or cross-channel sensor laws (Wang et al., 24 Apr 2025, Li et al., 2023, Mayo et al., 29 Jun 2025, Rajput et al., 12 Apr 2026, Halder et al., 31 Jul 2025, Zhang et al., 2023).

This class of methods differs from classical filtering in two respects. First, the physical prior is usually differentiable and is optimized jointly with the denoiser. Second, the prior need not be limited to smoothness or frequency heuristics; it may express Kirchhoff’s laws, PDE residuals, acquisition operators, or forward-model consistency. It also differs from purely data-driven deep denoising because clean labels are not always required. Some frameworks train from paired synthetic data, some from imperfect labels, some from unpaired simulated and experimental domains, and some in zero-shot or label-free settings (Awan et al., 2024, Cui et al., 19 Nov 2025, Xie et al., 2024, Asgariandehkordi et al., 26 Jun 2025, Gao et al., 2020).

A plausible implication is that “physics-informed” should be understood operationally rather than architecturally. The essential property is the insertion of a physically meaningful constraint into the denoising pipeline, not the use of any one backbone such as a U-Net, a PINN, or a DDPM.

2. Mathematical formulations

A recurring formulation starts from a standard denoising or diffusion objective and augments it with a physical residual. In the power-flow DDPM, the forward process is

$$q(x_t|x_{t-1})=\mathcal N(x_t;\sqrt{1-\beta_t}\,x_{t-1},\beta_t I),$$

with marginal

$$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$

and training begins from the usual simplified DDPM objective

$$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$

Physics is then inserted through the power-flow imbalance residual $R(x_t)$ and the step-wise penalty

$L_R(x_t)=\max\{R(x_t)-\gamma_t,0\},$

yielding

$L_{total}=L_{VLB}+\eta\,E_{t,x_t}[L_R(x_t)].$

The same work adds an auxiliary schedule-learning objective

$$L_{aux}(\omega)=\sum_{t=1}^T |R(x_t)-(t/T)\cdot\gamma_T|^2$$

to learn a forward-noise schedule aligned with physical imbalance growth (Wang et al., 24 Apr 2025).

More general diffusion formulations use PDE-style residual likelihoods. Physics-Informed Diffusion Models add a residual penalty of the form

$\frac{1}{2\bar\Sigma_t}\|\mathcal R(x_0^*)\|^2$

to the denoising loss, where $\mathcal R(\cdot)$ stacks interior and boundary residuals of the governing equations. PILD replaces the Gaussian-style virtual residual likelihood with a Laplace model

$$q_R(f|x)=\operatorname{Laplace}(f;R(x),b_t I),\qquad f=0,$$

so the negative log-likelihood becomes an $$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$0 penalty on $$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$1, combined with a gate $$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$2 that increases physics emphasis near the end of the reverse process (Bastek et al., 2024, Zeng et al., 29 Jan 2026).

In operator-based inverse problems, physics can enter the reverse step itself. PI-DDPM for microscopy shifts the reverse Gaussian mean by a gradient term proportional to

$$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$3

where $$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$4 is the imaging operator induced by the PSF. MRF-DiPh for multiparametric MRI uses a Half-Quadratic Splitting scheme in which each diffusion step alternates between a DDPM denoising proximal map and a physics proximal step that enforces both k-space data fidelity and the Bloch constraint set $$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$5 (Li et al., 2023, Mayo et al., 29 Jun 2025).

Outside diffusion, the same principle appears as multi-term denoising losses. PINNED minimizes

$$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$6

where the extra terms preserve smooth regions, Fourier-domain lattice periodicity, and physically meaningful brightness and contrast. PILOT, designed for noisy sensor windows without clean targets, minimizes

$$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$7

with $$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$8 enforcing algebraic or differential relations between channels. The zero-shot ultrasound method uses a residual self-supervised loss

$$q(x_t|x_0)=\mathcal N(x_t;\sqrt{\bar\alpha_t}\,x_0,(1-\bar\alpha_t)I),$$9

where the training pairs are two half-compounded images that share the same tissue response but contain different incoherent artifacts (Awan et al., 2024, Zhang et al., 2023, Asgariandehkordi et al., 26 Jun 2025).

3. Modes of physical integration

The literature exhibits several distinct insertion points for physics. The first is loss-level regularization. Examples include power-imbalance bounds in smart-grid diffusion, PDE residual penalties in diffusion models, TV and spectral-fidelity terms in STEM denoising, and PINN-guided consistency losses in additive manufacturing data denoising (Wang et al., 24 Apr 2025, Bastek et al., 2024, Awan et al., 2024, Halder et al., 31 Jul 2025).

The second is process-level guidance, where physics alters the denoising dynamics rather than merely regularizing the final output. PI-DDPM perturbs the reverse diffusion mean using the microscope forward model; MRF-DiPh inserts ADMM-based measurement and Bloch-consistency steps into every selected diffusion iteration; PAUFNO uses a warmup-weighted FK loss to suppress residual coupling-noise energy in the targeted Fourier cones of DAS data with imperfect labels (Li et al., 2023, Mayo et al., 29 Jun 2025, Cui et al., 19 Nov 2025).

The third is conditioning and architectural embedding. PILD injects physical or observational data via U-FiLM or U-Att at multiple U-Net layers. The net-load diffusion model embeds precomputed PV basis curves and condition vectors into a temporal U-Net so that the denoiser jointly learns diffusion and differentiable physics parameters. RMDM uses a dual U-Net design in which the first network enforces Helmholtz-based PINN constraints and the second performs diffusion-based denoising refinement (Zeng et al., 29 Jan 2026, Zhang et al., 2024, Jia et al., 31 Jan 2025).

The fourth is physics-informed data construction or self-supervision. TIE phase-map denoising generates paired $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$0 examples by simulating the full TIE imaging chain and inverse Laplacian reconstruction. PDA-Net creates simulated clean and blurry STM images from a Green’s-function LDOS forward model and then transfers the learned mapping to real experimental images through adversarial domain adaptation. The ultrasound zero-shot method obtains self-supervised pairs by partitioning transmission angles into two disjoint subsets, exploiting the assumption that tissue echoes are coherent across angles while angle-dependent artifacts are not (Rajput et al., 12 Apr 2026, Xie et al., 2024, Asgariandehkordi et al., 26 Jun 2025).

These mechanisms are complementary rather than exclusive. Several systems combine more than one: synthetic forward-model data generation plus learned denoising, or diffusion priors plus explicit physics projections. This suggests that the framework is modular by construction.

4. Representative architectures and workflows

The term covers a heterogeneous but structurally recognizable set of pipelines. Representative instances are summarized below (Awan et al., 2024, Li et al., 2023, Wang et al., 24 Apr 2025, Asgariandehkordi et al., 26 Jun 2025, Mayo et al., 29 Jun 2025, Zhang et al., 2023, Halder et al., 31 Jul 2025).

Setting	Architecture or workflow	Physical mechanism
Atomic-resolution STEM	Encoder–decoder CNN with three-conv blocks	TV, spectral fidelity, brightness, contrast losses
Light microscopy	Conditioned DDPM with 2D U-Net	PSF-based reverse-step guidance
Power-flow synthesis	U-Net-style DDPM with schedule network	AC power-flow imbalance penalty and learned $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$1
Low-angle ultrasound CPWC	Two-layer CNN trained zero-shot	Angle-subset self-supervision from coherence physics
Accelerated MRI fingerprinting	Pretrained DDPM plus HQS/ADMM loop	k-space consistency and Bloch constraint
Real-life sensing	1D convolutional autoencoder	Soft algebraic/differential sensor-law penalty
LPBF thermal data	FFNN denoiser with frozen PINN surrogate	PINN consistency plus EBM or Fisher regularization

Despite their diversity, these workflows usually follow a common sequence. A base denoiser is selected for the statistical inverse task; a physical residual, forward model, or surrogate is defined; the two are coupled by a composite objective or guided update; and evaluation is performed with both standard image/signal metrics and physics-specific diagnostics. In some cases the denoiser is trained conventionally after physics-aware data generation; in others, the denoiser never sees a clean label and is instead guided by physics alone or by cross-view consistency.

5. Empirical behavior across application domains

Quantitative evidence in the surveyed papers indicates that physics-aware coupling is most visible when plain data-driven denoising is prone to infeasible outputs, hallucinations, or domain shift. In power-flow synthesis, average power imbalance on 5,000 generated samples falls from 0.25 and 0.34 for an uninformed DDPM to 0.013 and 0.017 for the proposed learned-schedule physics-informed DDPM on the IEEE 14-bus and 30-bus systems. Under out-of-distribution load perturbations to $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$2, the same model maintains imbalance below $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$3 p.u. PI-DDPM in microscopy reports small but consistent gains over a standard conditioned DDPM on BioSR, with PSNR increasing from 23.70 to 23.97, MS-SSIM from 0.784 to 0.795, and NRMSE decreasing from 0.070 to 0.069; the paper also attributes fewer hallucinated structures and better filament continuity to the physics term (Wang et al., 24 Apr 2025, Li et al., 2023).

For paired image denoising, PINNED reports average training-set PSNR/SSIM of 32.4/0.95 at noise level L1 and 22.8/0.71 at L5, while stating PSNR gains of 2–4 dB and SSIM improvements of 0.05–0.15 over vanilla MSE-only training. The TIE phase-map framework, trained only on synthetic paired data, generalizes zero-shot to real 25,000 fps recordings and reports mean jet-region gradient magnitude increasing from $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$4 to $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$5, signal-to-background ratio from $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$6 to $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$7, and background-noise standard deviation from $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$8 to $$L_{VLB}=E_{x_0,\epsilon,t}\bigl[\|\epsilon-\epsilon_\theta(\sqrt{\bar\alpha_t}x_0+\sqrt{1-\bar\alpha_t}\epsilon,t)\|^2\bigr].$$9 across 20 frames (Awan et al., 2024, Rajput et al., 12 Apr 2026).

In zero-shot ultrasound plane-wave denoising, the proposed method exceeds BM3D, ZS-N2N, and supervised RF baselines on simulation, phantom, and in vivo carotid data. On the PICMUS simulation, for example, Row 3 improves from 0.80/3.6 dB in the noisy 5-angle input to 0.99/5.4 dB in gCNR/CNR, approaching the 75-angle reference at 0.99/6.1 dB. In real-life sensing, PILOT reports real-time execution of approximately 4 ms for a 1 s window on a Raspberry Pi 4, while in inertial navigation the physics MSE drops from approximately 118.7 to 1.87 for acceleration and from 0.3565 to 0.038 for angular velocity; the same paper reports improved CO$R(x_t)$0 and HVAC reconstruction relative to classical and deep-learning baselines (Asgariandehkordi et al., 26 Jun 2025, Zhang et al., 2023).

In iterative inverse problems, MRF-DiPh reports lower error than the best baseline MRF-IDDPM, with T1 MAPE improving from 8.45% to 6.75%, T2 MAPE from 22.54% to 18.40%, TSMI NRMSE from 27.26 to 18.65, and k-space NRMSE from 36.06 to 22.82. For denoising with imperfect labels, PAUFNO reports improvements of 5–10% over UFNO and 10–20% over U-Net across RMSE, $R(x_t)$1SNR, and local similarity, while requiring approximately 372k parameters and inferring a $R(x_t)$2 field in 0.39 s. In additive manufacturing, the denoiser guided by a frozen PINN surrogate reports on real experimental TEP data a reduction of RMSE from 532.5 for a vanilla model to 491.5 with EBM regularization and 462.3 with Fisher regularization, along with SNR increasing from 12.56 dB to 13.10 dB and 13.09 dB, respectively (Mayo et al., 29 Jun 2025, Cui et al., 19 Nov 2025, Halder et al., 31 Jul 2025).

Taken together, these results suggest that physics-informed denoising is especially effective when the corruption model overlaps structurally with the signal, when measurements are sparse, or when naive denoisers can satisfy pixelwise criteria while violating domain constraints.

6. Limitations, misconceptions, and research directions

A common misconception is that physics-informed denoising guarantees exact physical validity. The surveyed papers do not support that interpretation. Several explicitly note dependence on model fidelity: PINNED is trained solely on synthetic CDN noise and may not capture real correlated STEM artifacts; PILOT assumes a reasonably accurate differentiable law $R(x_t)$3 and warns that misspecification can inject bias; the additive-manufacturing framework freezes the PINN surrogate, so any surrogate error transfers directly to the denoiser (Awan et al., 2024, Zhang et al., 2023, Halder et al., 31 Jul 2025).

A second misconception is that such methods always require clean labels. Multiple counterexamples appear in the literature. PILOT trains without clean ground truth by combining reconstruction-to-noisy-input pretraining with a physics penalty. The ultrasound method is zero-shot and image-specific, requiring no separate dataset. PINN-SR performs super-resolution and denoising of fluid flow without any high-resolution labels, using only conservation laws and boundary conditions. PDA-Net learns from unpaired simulated clean/blurry STM images and real experimental images via adversarial domain adaptation (Zhang et al., 2023, Asgariandehkordi et al., 26 Jun 2025, Gao et al., 2020, Xie et al., 2024).

A third misconception is that physics must be enforced explicitly at inference. Some frameworks do so, but others do not. PIDM states that no extra physics term is needed at inference because the model has already internalized the PDE constraints during training; PILD likewise performs ordinary DDIM sampling conditioned on $R(x_t)$4 after training. By contrast, PI-DDPM and MRF-DiPh integrate physics directly into the reverse or optimization steps during sampling or reconstruction (Bastek et al., 2024, Zeng et al., 29 Jan 2026, Li et al., 2023, Mayo et al., 29 Jun 2025).

Open problems reported in the surveyed work are relatively consistent. They include hand-tuned loss weights, synthetic-only corruption models, the computational burden of iterative samplers or dictionary matching, and the difficulty of extending current formulations to richer or more weakly specified physics. The cited papers propose, or imply, several directions: automatic weight selection in multi-term losses, more expressive closure models for residual dynamics, differentiable replacements for costly projection steps, better handling of correlated or non-Gaussian noise, and faster inference through distillation or reduced-step samplers (Awan et al., 2024, Zeng et al., 29 Jan 2026, Mayo et al., 29 Jun 2025, Asgariandehkordi et al., 26 Jun 2025).

Within this literature, the central idea remains stable: denoising performance is improved not by replacing statistical learning with first principles, but by constraining learned denoisers so that the recovered signal lies in a physically meaningful subset of the ambient data space.