Physics-Informed Denoising Network (PIDN)

• PIDN is a denoising approach that integrates physics constraints to define plausible clean signals from measurement processes and governing equations.
• It employs mechanisms such as self-supervised acquisition-aware pairing, explicit physics-based regularization, and forward-model guided diffusion in network architectures.
• PIDN has been successfully applied in ultrasound, STEM, and flow imaging, achieving high noise reduction while preserving critical signal characteristics.

Physics-Informed Denoising Network (PIDN) denotes a class of denoising models in which the reconstruction map is constrained by known properties of the measurement process, governing equations, acquisition geometry, or physically meaningful signal statistics. In the recent literature, the label covers lightweight zero-shot residual CNNs for low-angle ultrasound coherent plane wave compounding, encoder–decoder denoisers for atomic-resolution STEM, self-supervised sensor denoisers regularized by algebraic physics, diffusion models with explicit imaging operators, and coordinate-based PINNs that reconstruct clean flow fields from noisy observations (Asgariandehkordi et al., 26 Jun 2025, Awan et al., 2024, Zhang et al., 2023, Li et al., 2023, Zhang et al., 15 Aug 2025). The term therefore refers less to a single network family than to a design principle: denoising is shaped by physics through the loss, the sampling strategy, the architecture, the synthetic training pipeline, or an external physics surrogate.

1. Conceptual scope and definition

A PIDN is typically built around a denoiser $f_\theta$ or an equivalent score/noise predictor, but differs from a purely data-driven denoiser in that physical knowledge explicitly constrains what counts as a plausible clean signal. Depending on the domain, that knowledge may be a forward imaging model, a PDE or ODE, cross-sensor algebraic relationships, conservation rules, acquisition diversity, or physically meaningful global statistics such as spectral content, brightness, and contrast (Awan et al., 2024, Zhang et al., 2023).

This category is broader than the classical PDE-focused notion of a Physics-Informed Neural Network. In atomic-resolution STEM, the model called PINNED is explicitly described as not physics-informed in the PINN-for-PDE sense; instead, physics enters through spectral fidelity, total variation, and brightness/contrast consistency losses, together with synthetic data generated from crystallographic structure and a composite noise model (Awan et al., 2024). At the other extreme, evidential physics-informed formulations treat denoising as recovery of a latent mean field $\gamma$ constrained by a PDE residual, while predictive uncertainty is represented through normal-inverse-gamma hyperparameters with variance

$$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$

so that denoising and uncertainty quantification are solved jointly (Tan et al., 27 Jan 2025).

A plausible implication is that PIDN is best understood as a continuum. Some instances are strongly model-based, with differential operators enforced by automatic differentiation; others are weakly physics-informed, using acquisition-aware pair construction, physics-derived embeddings, or architectural conservation constraints. What unifies them is that denoising is not learned from appearance statistics alone.

2. Mechanisms by which physics enters the denoiser

One common mechanism is acquisition-aware self-supervision. In low-angle CPWC ultrasound, the available steering angles $\{P_1,\dots,P_k\}$ are split into two disjoint subsets, yielding two compounded images that share the same underlying anatomy but differ in angle-dependent artifacts. The paper models these as

$$S_1(y)(\mathbf{r}) = s(\mathbf{r}) + n_1(\mathbf{r}), \qquad S_2(y)(\mathbf{r}) = s(\mathbf{r}) + n_2(\mathbf{r}),$$

and trains a residual denoiser through a symmetric loss

$$L_{\text{Res.}} = \frac{1}{2}\Big( \|S_1(y) - f_\theta(S_1(y)) - S_2(y)\|_1 + \|S_2(y) - f_\theta(S_2(y)) - S_1(y)\|_1 \Big),$$

augmented by a gradient consistency term and the residual output rule $\hat{x}=y-f_\theta(y)$ (Asgariandehkordi et al., 26 Jun 2025). Here the “physics” is angular diversity: tissue is approximately angle-coherent, while sidelobes, grating lobes, aberration effects, and incoherent noise vary with angle.

A second mechanism is explicit physics-aware regularization of the objective. In STEM, PINNED minimizes

$$\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{MSE}} + \lambda_{\text{TV}}\mathcal{L}_{\text{TV}} + \lambda_{\text{SF}}\mathcal{L}_{\text{SF}} + \lambda_B \mathcal{L}_B + \lambda_C \mathcal{L}_C,$$

where total variation suppresses noise in homogeneous regions, spectral fidelity preserves Fourier magnitude, and brightness/contrast consistency constrain physically meaningful intensity statistics (Awan et al., 2024). In real-life sensing systems, PILOT instead uses self-supervised reconstruction plus a physics penalty,

$$\ell(\theta)=\ell_{\mathrm{rec}}+\lambda\,\ell_{\mathrm{phy}},$$

with

$$\ell_{\mathrm{rec}}=\|f(\mathbf{Y}_i+\mathbf{n}_i;\theta)-\mathbf{Y}_i\|_2^2, \qquad \ell_{\mathrm{phy}}=\|g(f(\mathbf{Y}_i+\mathbf{n}_i;\theta))\|_2^2,$$

so that the denoised outputs satisfy kinematic, mass-balance, or heat-balance relations even though no clean target is available (Zhang et al., 2023).

A third mechanism is explicit incorporation of the forward operator into a generative denoiser. In microscopy reconstruction, PI-DDPM starts from the acquisition model

$\gamma$0

with PSF $\gamma$1, background $\gamma$2, and Poisson noise operator $\gamma$3, then adds the gradient of a data-fidelity term $\gamma$4 to the reverse diffusion dynamics. The reverse step is therefore not purely learned denoising; it is learned denoising plus a physics-based correction toward consistency with the optics (Li et al., 2023).

Across the literature, this suggests three recurrent PIDN primitives: signal-consistent/noise-inconsistent view construction, explicit physical penalties in the loss, and differentiable forward-model guidance during inference or sampling.

3. Architectures and training regimes

Architecture is highly variable, because the defining feature of PIDN lies in the constraints rather than the backbone. The ultrasound PIDN is deliberately minimal: two $\gamma$5 convolutional layers with 48 feature channels, Leaky ReLU activations, a final $\gamma$6 convolution, and fewer than 22k parameters, trained per image from scratch in about 1000 iterations (Asgariandehkordi et al., 26 Jun 2025). By contrast, PINNED for STEM is a deep encoder–decoder with multiple three-convolution blocks and transposed-convolution upsampling (Awan et al., 2024). PILOT uses a lightweight 1D CNN with four temporal convolution layers of kernel sizes $\gamma$7, optimized for deployment on embedded hardware (Zhang et al., 2023).

Diffusion-based and surrogate-based PIDNs are structurally different again. PI-DDPM adopts a U-Net-style noise-prediction network conditioned on the degraded observation and modified by a forward-model gradient (Li et al., 2023). For noisy variational quantum algorithms, PIDN becomes a dual-branch surrogate with a GRU history encoder, a feedforward branch for the current noisy cost, and two output heads for denoised cost and next-step parameters, trained to match ZNE-corrected trajectories and gradients (Liu et al., 3 May 2026). In noisy flow-image reconstruction, the denoising component is an 8-layer MLP PINN for velocity and pressure, coupled to a U-Net that predicts a quasi-conformal deformation field for domain correction (Zhang et al., 15 Aug 2025).

Domain	Physics injection	Representative implementation
Low-angle CPWC ultrasound	Angle-split self-supervision and residual consistency	Lightweight residual CNN, zero-shot, two conv layers (Asgariandehkordi et al., 26 Jun 2025)
Atomic-resolution STEM	Spectral, TV, brightness, and contrast constraints	Encoder–decoder PINNED (Awan et al., 2024)
Real-life sensor streams	Algebraic physics loss without clean labels	1D CNN PILOT (Zhang et al., 2023)
Microscopy reconstruction	PSF-based forward model in diffusion denoising	PI-DDPM with physics-guided reverse process (Li et al., 2023)
Flow images	Navier–Stokes residuals plus domain reconstruction	PINN + quasi-conformal U-Net (Zhang et al., 15 Aug 2025)
Variational quantum algorithms	Surrogate of ZNE optimization dynamics	GRU/FFNN PIDN (Liu et al., 3 May 2026)

Training regimes are equally diverse. Some PIDNs are zero-shot and per-instance, as in ultrasound. Others are supervised on synthetic data generated by a forward model, as in STEM and TIE phase-map denoising. Still others are self-supervised, with artificial corruption added to already noisy data and physics terms used to resolve the bias introduced by that strategy, as in PILOT.

4. Representative applications and empirical behavior

In ultrasound CPWC, the method targets low-angle acquisitions where few steering angles reduce frame-rate loss but leave substantial noise and artifact residuals. Evaluations on simulation, phantom, and in vivo data show that the proposed method is consistently best or among the best, often approaching the 75-angle reference. In the in vivo carotid longitudinal setting, the reported result is gCNR $\gamma$8 and CNR $\gamma$9, compared with $$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$0 and $$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$1 for 75-angle CPWC; in carotid cross-section, the proposed method reaches gCNR $$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$2 and CNR $$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$3, versus $$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$4 and $$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$5 for the 75-angle reference (Asgariandehkordi et al., 26 Jun 2025).

For transient-flow phase maps reconstructed by TIE, the denoiser is trained solely on a physics-informed synthetic dataset whose clean targets are procedurally generated compressible-flow morphologies passed through forward TIE simulation and inverse-Laplacian reconstruction. On real phase maps acquired at 25,000 fps, the reported gains are a 13,260% improvement in signal-to-background ratio and a 100.8% improvement in jet-region structural sharpness across 20 evaluated frames, while background-noise standard deviation is strongly reduced (Rajput et al., 12 Apr 2026).

In real-life sensing systems, PILOT is evaluated on inertial navigation, CO$$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$6 monitoring, and HVAC control. The paper reports state-of-the-art performance relative to Gaussian filtering, DWT, DnCNN, TSTNN, DIP, and Neighbor2Neighbor, while remaining deployable in real time: 4 ms for a sequence of 1 s on Raspberry Pi 4, with model size about 284 KB and CPU utilization around 25% (Zhang et al., 2023).

Physics-informed denoising also appears in domains not usually associated with image restoration. In LPBF additive manufacturing, a denoiser guided by a pretrained PINN surrogate and EBM/Fisher-score regularization is benchmarked on the simple harmonic oscillator, Burgers’ equation, Laplace’s equation, and thermal-emission data from LPBF; the guided denoisers outperform a vanilla feedforward network across noise levels and process conditions (Halder et al., 31 Jul 2025). In experimental stratified flows, a coordinate-based PINN is reported to eliminate measurement noise, correct scanning distortion, identify weak but dynamically important three-dimensional vortices, revise turbulent energy budgets and mixing efficiency, and predict the latent pressure field (Zhu et al., 2023). In noisy variational quantum algorithms, PIDN reproduces ZNE-mitigated optimization dynamics while reducing circuit executions by approximately a factor of 4 to 6 and maintaining gradient cosine similarity above 0.95 throughout training (Liu et al., 3 May 2026).

5. Relation to adjacent methods and recurrent misconceptions

A frequent misconception is that any self-supervised denoiser is already physics-informed. The ultrasound literature makes the distinction explicit: Noise2Noise uses independently noisy pairs, while Noise2Void, Noise2Self, and Neighbor2Neighbor exploit masked pixels or spatial neighbors, and Zero-Shot Noise2Noise uses spatial downsampling. The CPWC PIDN replaces generic spatial manipulations with angle-based pseudo-pairs derived from acquisition physics; in the same study it is compared against BM3D, MSLAE of Perdios et al., DPWDPM of Asgariandehkordi et al., and ZS-N2N of Mansour et al. (Asgariandehkordi et al., 26 Jun 2025).

Another misconception is that physics-informed denoising must operate directly on the measurements. Post hoc unlearning and pruning offer a different route. P-PINN begins with a pretrained PINN, partitions data into reliable and corrupted subsets using a joint residual–data fidelity indicator, computes a bias-based neuron-importance score, prunes noise-sensitive neurons iteratively, and fine-tunes on the retained data under the original PDE constraints. The denoising mechanism here acts on internal representations rather than directly on pixels or sensor streams (Chen et al., 23 Feb 2026).

A third ambiguity concerns the boundary between “physics-informed” and “physics-inspired.” WIPUNet for image denoising adopts conservation, locality, isolation, and conditioning principles borrowed from pileup mitigation in high-energy physics, enforcing residual subtraction of the form $$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$7 or $$\sigma_p^2 = \frac{\beta}{\alpha - 1}\left(1 + \frac{1}{\nu}\right),$$8. This suggests that some works use hard architectural inductive biases rather than explicit forward-model residuals, yet still belong to the broader PIDN family insofar as physical structure constrains what the denoiser may predict (Islam, 6 Sep 2025).

6. Limitations, open problems, and likely directions

The chief limitation of any PIDN is the validity and usefulness of the injected physics. In ultrasound, the zero-shot CPWC method requires multiple steering angles, is sensitive to motion between transmissions, and still incurs per-image training cost on the order of seconds; in TIE phase-map denoising, synthetic training data simplify flow morphologies, background structure, and nozzle geometry, which leads to residual mismatch on real data (Asgariandehkordi et al., 26 Jun 2025, Rajput et al., 12 Apr 2026). In physics-guided LPBF denoising, performance depends on the quality of the PINN surrogate and remains constrained by limited high-resolution experimental data (Halder et al., 31 Jul 2025).

PINN-based PIDNs add their own numerical burdens. Flow-image denoising with Navier–Stokes constraints is sensitive to loss balancing, collocation, and the interaction between field reconstruction and domain reconstruction; PDE-discovery dPINNs similarly depend on DFT thresholds, sparse-regression hyperparameters, and multi-stage optimization choices (Zhang et al., 15 Aug 2025, Thanasutives et al., 2022). These issues are not incidental: they reflect the general difficulty of optimizing neural networks under stiff, multi-objective physical constraints.

The literature nevertheless indicates several stable directions for extension. One is hybridization between per-instance adaptation and reusable priors, such as the “universal prior” plus per-image fine-tuning proposed for ultrasound (Asgariandehkordi et al., 26 Jun 2025). Another is stronger and more explicit incorporation of domain physics, such as richer STEM image-formation models or more detailed detector response (Awan et al., 2024). A third is uncertainty-aware PIDN design, exemplified by evidential formulations that treat the denoised mean field and its uncertainty jointly (Tan et al., 27 Jan 2025). A fourth is broader multimodal or online deployment, including multi-sensor LPBF monitoring, online adaptation in sensing systems, and end-to-end TIE-consistency losses for phase-map reconstruction (Halder et al., 31 Jul 2025, Zhang et al., 2023, Rajput et al., 12 Apr 2026).

Taken together, these works indicate that PIDN is not a narrow method but a research program. Its central claim is that denoising improves when the network is told not merely what clean data look like, but what clean data are allowed to be under the physics of their generation, transmission, or downstream use.