Physics-Informed Noise Model

• Physics-informed noise models integrate governing physical laws into machine learning to address non-Gaussian, signal-dependent noise.
• They employ hybrid loss functions that combine data fidelity with physics residuals and energy-based probabilistic frameworks for robust noise handling.
• Applications include PDE discovery, sensor data denoising, quantum system identification, and control, achieving significant error reductions even under extreme noise conditions.

Physics-Informed Noise Model

A physics-informed noise model leverages known physical constraints, governing equations, or physically-derived structural assumptions to regularize, represent, or directly model noise and uncertainty in scientific machine learning. This paradigm has proven essential across domains including PDE discovery, sensor data denoising, quantum system identification, image formation, and control, delivering robustness under measurement noise and capacity for interpretable uncertainty quantification. Core instantiations include hybrid loss functions combining data fidelity with automatic differentiation-based physics residuals, explicit probabilistic modeling of measurement or input noise, energy-based noise density learning, and architectures that enforce physical invariants or symmetries. The approach is varied, encompassing both deterministic constraints and fully probabilistic Bayesian treatments, and can directly address non-Gaussian, heteroskedastic, or signal-dependent noise encountered in real scientific and engineering problems.

1. Foundational Principles

Early neural surrogate modeling suffered acute degradation under noise due to lack of physical priors, particularly when dealing with sparse or ill-posed data. The physics-informed approach integrates domain knowledge, typically in the form of governing PDEs, conservation laws, or analytic relationships, to regularize the learning process. For example, the Physics-Informed Neural Network (PINN) framework utilizes composite loss functions where classic data-matching terms—$$\mathcal{L}_{\rm data} = \sum_i \|\hat y(x_i) - y^\mathrm{noisy}_i\|^2$$—are augmented by physics-mismatch residuals evaluated via automatic differentiation, such as $$\mathcal{L}_{\rm phys} = \sum_j [R_{\rm PDE}(x_j)]^2$$ (Wong et al., 2021). This structure systematically filters out signal perturbations that violate the underlying physics, whether the noise is Gaussian, non-Gaussian, or structure-dependent.

2. Energy-based and Probabilistic Noise Modeling

To overcome limitations of simple $\ell_2$ data fitting under non-Gaussian or unknown noise, energy-based models (EBMs) and probabilistic frameworks have been embedded into the PINN training process. For instance, the PINN-EBM method jointly trains the solution net and an energy-based noise density $$p_\theta(\epsilon) = e^{-E(\epsilon;\theta_n)}/Z(\theta_n)$$, with the EBM loss replacing least-squares: $$\mathcal{L}_{\rm data}^{\rm EBM} = -\frac{1}{N_d} \sum_{i=1}^{N_d} \log p_{\theta_n}(r_i) = \frac{1}{N_d} \sum_{i=1}^{N_d} \left[ E(r_i;\theta_n) + \log Z(\theta_n) \right]$$ where $r_i = y_i - u(x_i;\theta_p)$ are the residuals (Pilar et al., 2022). This adaptation allows accurate parameter recovery even under arbitrary non-Gaussian distributions—heavy-tailed, multimodal, or skewed—and yields unbiased solutions guided by the learned physics and the empirical noise statistics.

3. Physics-Informed Denoising Architectures

Denoising networks combining structural priors with physical constraints have shown notable efficacy in sensor fusion, additive manufacturing, microscopy, and field interpolation. For time-series and multi-channel sensor data, approaches such as PILOT utilize a lightweight 1D-CNN autoencoder trained via a two-term loss: $$\mathcal{L}(\theta) = \mathcal{L}_{\rm rec}(\theta) + \lambda \mathcal{L}_{\rm phy}(\theta)$$ with $\mathcal{L}_{\rm phy}$ encoding analytic physics constraints (e.g., Newton’s second law, mass-balance, or kinematic relationships) (Zhang et al., 2023). The absence of ground-truth clean data is alleviated by self-supervised denoising with injected artificial noise and “soft” physics penalties. In image domains, physics-informed noise neural proxies (PNNP) are trained solely on dark-frame calibrations, with signal-dependent components inferred via physics-guided decoupling and distribution-matching losses (CDF/quantile) (Feng et al., 2023). This strategy avoids the need for paired data and addresses realistic sensor noise complexity.

4. Quantum and Stochastic System Identification

In open quantum systems, physics-informed noise modeling is realized by parameterizing both Hamiltonian and dissipation parameters in neural networks subject to Lindblad master equation constraints. For instance, PINNverse applies

$$\frac{d\rho(t)}{dt} = -i[H(\theta_H),\rho(t)] + \sum_k \gamma_k (L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho\})$$

and treats $\{J_{\mu,\nu}\}$ and $\{\gamma_k\}$ as trainable variables refined on noisy data via composite losses integrating physics residuals and measurement fits (Lima et al., 16 Jul 2025). This yields robust Hamiltonian and noise parameter identification from few noisy observations, with sample efficiency far exceeding standard tomographic approaches. In stochastic dynamical systems, Kramers’-inspired barrier losses and SDE residuals quantitatively regularize the surrogate learning process to accurately predict noise-induced phenomena (e.g. self-induced stochastic resonance) (Savaliya et al., 26 Oct 2025).

5. Noise-Robust Automatic PDE Discovery

Physics-informed noise models underpin robust PDE identification via multi-stage frameworks such as nPIML, which orchestrates

1. supervised learning of solution and derivative estimates,
2. unsupervised self-gating preselection of candidate basis functions,
3. sparse regression for initial PDE selection, and
4. denoising PINN fine-tuning using DFT-based projection networks. Explicit spectral denoising and learnable additive corrections act on the feature library and measurement data, enabling recovery of parsimonious governing equations from data corrupted by arbitrary (not necessarily Gaussian) noise (Thanasutives et al., 2022). The approach achieves lowest coefficient errors across canonical PDEs compared to classical regression and vanilla PINNs, especially under input/output noise.

6. Benchmark Results and Domain-Specific Performance

Numerical evidence across domains repeatedly confirms the noise-robustness of physics-informed noise models. For fluid dynamics surrogates, inclusion of Navier–Stokes-based residuals yields up to 10× lower test error under 20% Gaussian noise (Wong et al., 2021). PINN-EBM achieves best-in-class parameter recovery under biased, multimodal noise. PILOT and physics-guided denoiser networks outperform vanilla neural architectures and statistical prefilters in real sensor and manufacturing data, with up to two orders of magnitude reduction in physics constraint violations and substantial SNR increases (Zhang et al., 2023, Halder et al., 31 Jul 2025). Quantum PINNs recover correct dissipation rates and Hamiltonian structure from sparse, noisy measurement records with scaling advantages for multi-qubit devices (Lima et al., 16 Jul 2025, Sulc, 15 Sep 2025). Physics-informed DDPMs in microscopy prevent hallucinated structures and minimize reconstruction error compared to standard deep learning and deconvolution methods (Li et al., 2023).

Approach	Noise Type Addressed	Key Physics Constraint
PINN-EBM (Pilar et al., 2022)	Non-Gaussian, Unknown	PDE residual via autodiff
PILOT (Zhang et al., 2023)	Real sensor, non-Gaussian	Analytic algebraic constraint
PNNP (Feng et al., 2023)	Sensor (dark), signal-dep.	Physical noise decoupling
PINNverse (Lima et al., 16 Jul 2025)	Gaussian (quantum meas.)	Lindblad master equation
nPIML (Thanasutives et al., 2022)	Arbitrary (input/output)	Sparse regression + DFT denoise

7. Limitations, Interpretability, and Future Directions

While physics-informed noise models have proven highly robust, several limitations persist:

• Sensitivity to the appropriateness of the physics constraint; incorrect or oversimplified priors may induce bias.
• The need for explicit denoising or regularization in the presence of extreme noise.
• Ad hoc hyperparameter scheduling (e.g. thresholding in DFT denoising) that may require domain adaptation (Thanasutives et al., 2022).
• Extension to non-polynomial or nonlocal PDE structures is an open area.

Interpretability is improved via feature-importance rankings, explicit symmetry or conservation law enforcement, and well-defined probabilistic noise models (energy-based or DDL-based). Promising directions include integration of non-Markovian or time-dependent noise processes, heterogeneous or structured noise in multi-modal sensor systems, and acceleration for ultra-low-latency real-time applications.

Physics-informed noise modeling now stands as a central instrument for marrying theory-guided regularization, sample-efficient learning, and robust generalization under realistic experimental conditions.