Physics-Informed Diffusion Models

• Physics-informed diffusion models are diffusion-based frameworks that incorporate physical laws and measurement operators to ensure generated outputs abide by governing equations.
• They integrate physics through training losses, inference guidance, and architectural constraints, significantly reducing residual errors in applications like fluid dynamics and cardiac CT.
• Applied across diverse domains, these models enhance data fidelity and computational efficiency, offering practical benefits in simulation, reconstruction, and anomaly detection.

Physics-informed diffusion models are diffusion-based generative, reconstruction, and surrogate-learning frameworks in which physical laws, physical priors, or measurement operators are incorporated into the diffusion process so that generated samples, reconstructions, or forecasts remain aligned with governing equations, boundary conditions, kinematic constraints, or domain-specific forward models. In the literature summarized here, the term spans denoising diffusion probabilistic models, score-based diffusion models, latent diffusion models, and function-space or spectral variants, with physics entering through training losses, inference-time guidance, architectural constraints, reward formulations, and simulator-derived conditioning. The resulting systems have been applied to PDE benchmarks, fluid dynamics, cardiac CT, microscopy, infrared imaging, radio-map reconstruction, air-pollution forecasting, net-load generation, vehicle trajectory synthesis, and trajectory anomaly detection (Bastek et al., 2024, Yuan et al., 24 Sep 2025, Wang et al., 9 Jun 2025).

1. Scope and design space

A physics-informed diffusion model is not a single algorithmic template. The papers considered here describe a design space in which the diffusion prior is combined with first-principles structure at different loci of the pipeline.

Integration locus	Typical mechanism	Representative papers
Training objective	PDE residual penalties, virtual residual likelihoods, weighted physics losses	(Bastek et al., 2024, Zeng et al., 29 Jan 2026, Soni et al., 15 Aug 2025)
Sampling and inference	Data-consistency gradients, PDHG, posterior guidance, residual correction	(Li et al., 2023, Han et al., 2024, Shi et al., 2024)
Architecture and representation	Stream-function decoders, DeepONet, spectral latent spaces, dual U-Nets	(Wang et al., 9 Jun 2025, Zhou et al., 4 Dec 2025, Gallon et al., 10 Feb 2026, Jia et al., 31 Jan 2025)
Decision-theoretic formulation	Terminal reward optimization over denoising trajectories	(Yuan et al., 24 Sep 2025)

This range is important because the phrase “physics-informed” can denote materially different mechanisms. In some works, the model is trained to internalize physical residual minimization directly. In others, the pretrained diffusion prior is corrected at test time by gradients of a measurement model or PDE residual. Still other works encode invariances or conservation laws by construction, such as divergence-free decoders or symmetry-enforcing decoders. A plausible implication is that the field should be understood less as a single method family than as a family of interfaces between diffusion priors and scientific structure (Wang et al., 9 Jun 2025, Gallon et al., 10 Feb 2026).

2. Training-time incorporation of physics

A central line of work modifies the diffusion training objective so that denoising and physical consistency are optimized jointly. In “Physics-Informed Diffusion Models” the augmented objective is written as

$$L_{\text{PIDM}}(\theta)=\mathbb{E}\left[\lambda_t \|x_0-\hat{x}_0(x_t,t)\|^2+\frac{1}{2\bar{\Sigma}_t}\|\mathcal{R}(x_0^*(x_t,t))\|^2\right],$$

where $\mathcal{R}(\cdot)$ is the residual of the governing equations or boundary conditions. The paper presents this as a first-principle-based loss term that enforces generated samples to fulfill underlying physical constraints and reports that the residual error is reduced by up to two orders of magnitude in a fluid-flow case study, while also acting as a natural regularization mechanism against overfitting (Bastek et al., 2024).

“PILD: Physics-Informed Learning via Diffusion” replaces Gaussian virtual residual modeling with a Laplace residual observation,

$$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$

and optimizes a joint loss of the form

$$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$

Here the residual is evaluated on a DDIM-generated sample $x^*$, the weighting $A_t$ uses a Min-SNR schedule, and the gating function $G(t)$ increases the physics penalty as the diffusion state becomes cleaner. The framework is stated to support ODEs, PDEs, algebraic equations, and inequality constraints (Zeng et al., 29 Jan 2026).

Other training-time formulations are more domain-specific. “Pi-fusion” combines supervised data loss, Navier–Stokes residual loss, and diffusion noise-prediction loss in a single objective for learning the temporal evolution of velocity and pressure fields (Qiu et al., 2024). The multivariate time-series anomaly-detection model TPIDM adds a weighted physics-informed term

$$\mathcal{L}_{TPIDM}=\mathcal{L}_{DM}+\mathcal{L}_{PI},$$

where the static schedule $\overline{\lambda}_{PI_t}$ emphasizes early, less noisy diffusion steps because derivative-based residuals become unreliable at late noisy steps (Soni et al., 15 Aug 2025). In synthetic net-load generation, the signal is decomposed as $\mathbf S=\mathbf S_d+\mathbf S_p$, with a physics-informed solar-PV component embedded inside the denoising network and trained jointly with the diffusion transition kernel (Zhang et al., 2024). In infrared image generation, the latent diffusion backbone is regularized by a physical reconstruction loss and a TeV-space loss derived from a frozen decomposition network, without increasing training parameters (Mao et al., 2024).

Taken together, these works present physics-informed training not merely as regularization in the generic machine-learning sense, but as a way of reshaping the learned data distribution so that denoising trajectories terminate in physically admissible regions of state space.

3. Inference-time guidance, posterior correction, and data consistency

A second major tradition keeps the diffusion prior but enforces physics during sampling. In microscopy reconstruction, PI-DDPM incorporates the microscope image-formation model into both loss and reverse process. The reverse update includes the gradient of a data-fidelity term $\mathcal{R}(\cdot)$0, so that each denoising step is corrected toward agreement with the point-spread-function-based forward model. The reported effect is improved reconstruction quality together with artefact and hallucination reduction (Li et al., 2023).

For limited-angle cardiac CT, PSDM combines a score-based diffusion prior with the primal-dual hybrid gradient algorithm and Fourier fusion. The posterior score is decomposed into a learned image prior and a data-likelihood term,

$\mathcal{R}(\cdot)$1

and the reconstruction alternates between denoising, Fourier-domain fusion with a limited-angle reconstruction, and PDHG-based data-consistency updates (Han et al., 2024).

Fluid-dynamics reconstruction provides another inference-time pattern. The 2022 flow-field reconstruction model trains only on high-fidelity data, then at test time conditions sampling on low-fidelity or sparse measurements by constructing noisy guided states and, when available, adds PDE residual gradients as conditioning information or direct gradient-descent guidance. The same model is reported to reconstruct from regular low-fidelity samples and sparsely measured samples without retraining (Shu et al., 2022). “Self-Guided Diffusion Model for Accelerating Computational Fluid Dynamics” adds a residual-correction step during inference that minimizes Navier–Stokes residuals and combines it with a Predictor-Corrector-Advancer SDE solver, while a wavelet-based Importance Weight strategy focuses training on high-frequency flow details (Li et al., 6 Apr 2025).

Multi-fidelity surrogate modeling and spectral latent diffusion extend this posterior-correction view. DBS uses inexpensive simulations as conditioning during training and inference, then adds expensive simulations at inference through posterior guidance derived from the gradient of a simulation-based likelihood, all within a Bayesian probabilistic model equipped with a Wasserstein-distance guarantee (Shi et al., 2024). PISD performs guidance in a scaled spectral latent space, where observation and PDE gradients are applied at each reverse step through Adam-based updates, and reports improved accuracy and computational efficiency on Poisson, Helmholtz, and incompressible Navier–Stokes equations (Gallon et al., 10 Feb 2026).

This branch of the literature treats physics not as a fixed training regularizer but as an online constraint operator acting on the reverse process itself.

4. Reward formulations, function spaces, and representation choices

A notable reformulation appears in “PIRF: Physics-Informed Reward Fine-Tuning for Diffusion Models,” which casts unconditional diffusion sampling as a Markov Decision Process with terminal reward. The state is $\mathcal{R}(\cdot)$2, the action is $\mathcal{R}(\cdot)$3, the policy is the diffusion transition kernel, and the reward is defined only at the final sample:

$\mathcal{R}(\cdot)$4

Instead of relying on DPS-style value function approximations, PIRF backpropagates gradients of the terminal reward directly through full denoising trajectories,

$\mathcal{R}(\cdot)$5

The paper identifies a shared bottleneck in prior approaches based on diffusion posterior sampling–style value approximation and introduces layer-wise truncated backpropagation together with weight-based regularization to mitigate sample inefficiency and reward hacking (Yuan et al., 24 Sep 2025).

Function-space and spectral formulations address a different limitation: the mismatch between continuous physical quantities and pixel- or grid-based generative models. FunDiff combines a function autoencoder with latent diffusion so that inputs with varying discretizations can be encoded into a continuous latent space and decoded as continuous functions evaluable at arbitrary coordinates. Physical priors enter either as architectural constraints—periodicity, symmetry, divergence-free fields through a stream-function decoder—or as physics-informed loss terms such as PDE residual penalties. The paper also gives minimax density-estimation bounds in function spaces and states that diffusion-based estimators achieve optimal convergence rates under suitable regularity conditions (Wang et al., 9 Jun 2025).

PISD pushes this representation question into spectral space. By diffusing scaled spectral coefficients rather than grid values, it ensures that Gaussian perturbations correspond to functions with controlled regularity and that differential operators remain well defined throughout the process. This suggests that representation choice is not merely an efficiency issue; it determines whether physics operators are numerically meaningful during diffusion (Gallon et al., 10 Feb 2026).

A distinct but related usage of the phrase appears in “Physics Informed Distillation for Diffusion Models,” where the “physics” is the probability-flow ODE associated with a teacher diffusion model. The student network is trained by minimizing an ODE residual in a PINN-like manner, without synthetic trajectory generation during distillation (Tee et al., 2024). This usage broadens the term beyond physical science applications and shows that “physics-informed” can also refer to equation-constrained diffusion mechanics internal to the model itself.

5. Domains, architectures, and empirical patterns

The application range of physics-informed diffusion is unusually broad. In PDE-governed scientific generation, PIRF reports five benchmarks—Burgers, Darcy, Helmholtz, Poisson, and Kolmogorov—and states that it consistently achieves the lowest PDE residual MSE under 20, 40, and 80 sampling steps, with particularly large gains in low-step sampling regimes (Yuan et al., 24 Sep 2025). In radio-map reconstruction, RMDM uses a dual U-Net architecture in which a PINN-constrained first U-Net enforces Helmholtz-equation consistency and a second U-Net performs diffusion denoising; the reported results are NMSE $\mathcal{R}(\cdot)$6 and RMSE $\mathcal{R}(\cdot)$7 under the Static RM setting, and NMSE $\mathcal{R}(\cdot)$8 and RMSE $\mathcal{R}(\cdot)$9 under the Dynamic RM setting (Jia et al., 31 Jan 2025).

Imaging applications show a recurring emphasis on hallucination suppression. PI-DDPM for microscopy reports BioSR test performance of PSNR $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$0, MS-SSIM $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$1, and NRMSE $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$2, compared with PSNR $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$3, MS-SSIM $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$4, and NRMSE $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$5 for a regular conditioned DDPM, while also describing fewer artefacts and fewer hallucinations (Li et al., 2023). PSDM for limited-angle cardiac CT combines score-based priors and model-based updates for high-quality reconstruction from severely limited-angle data (Han et al., 2024). In infrared image translation, PID augments latent diffusion with TeV decomposition losses and reports FID improvement from $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$6 for the baseline LDM to $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$7 on KAIST, while emphasizing physically plausible temperature structure (Mao et al., 2024).

Time-dependent and sequential problems reveal a different set of design choices. STeP-Diff combines DeepONet with a PDE-informed diffusion model for mobile air-pollution forecasting and reports improvements of up to $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$8 in MAE, $$q_R(\tilde r \mid x_0)=\mathrm{Laplace}(\tilde r;R(x_0),\sigma I),$$9 in RMSE, and $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$0 in MAPE over the second-best algorithm after deployment of 59 portable sensors across two cities for 14 days (Zhou et al., 4 Dec 2025). For trajectory anomaly detection, Pi-DPM incorporates kinematic bicycle model regularization into an encoder–decoder diffusion framework and reports the highest anomaly-detection accuracy and F1 across Geolife, MarineCadastre, and Danish Maritime datasets, together with lower generation errors than VAE-, GAN-, and diffusion-based baselines (Sharma et al., 8 Jun 2025). For vehicle speed trajectory generation, a transformer-based Conditional Score-based Diffusion Imputation model with soft physics constraints achieves Wasserstein distance $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$1 for speed, $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$2 for acceleration, and discriminative score $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$3 on 6,367 GPS-derived micro-trips (Sokolov et al., 4 Feb 2026).

Other domains use physics-informed diffusion to replace expensive simulation or compensate for scarce data. Synthetic net-load generation embeds a solar-PV system performance model inside the denoising network and reports at least $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$4 improvement over state-of-the-art generative baselines across all quantitative metrics on Pecan Street data (Zhang et al., 2024). SAR ship-wake generation trains a latent diffusion model on images produced by a physics-based simulator and text prompts derived from simulation parameters; the generated Kelvin wake patterns are described as realistic, with significantly faster inference than the simulator and an approximately $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$5 speed difference for $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$6 images as reported in the detailed summary (Kamirul et al., 28 Apr 2025). In fluid dynamics, Pi-fusion reports RMSE $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$7 on synthetic 2D cylinder flow and substantially lower errors than PINN and NSFnets on 3D hepatic portal vein and brain artery data, together with inference around $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$8s per brain-artery sample versus $$\mathcal{L}_{\mathrm{PILD}}(\theta)=\mathbb{E}\left[A_t \|\epsilon-\epsilon_\theta(\sqrt{\alpha_t}x_0+\sqrt{1-\alpha_t}\epsilon,\mathcal O,t)\|_2^2 + G(t)\|R(x^*)\|_1\right].$$9s for numerical simulation (Qiu et al., 2024).

Across these domains, a shared empirical pattern is reported repeatedly: coupling diffusion priors with physically structured constraints improves physical consistency and often improves conventional fidelity metrics as well.

6. Misconceptions, limitations, and open directions

A common misconception is that “physics-informed” necessarily means hard constraint enforcement. The vehicle speed study explicitly reports catastrophic optimization failure for hard constraints in diffusion training, including $x^*$0 boundary violations and degraded distribution matching, whereas soft, threshold-activated penalties succeeded (Sokolov et al., 4 Feb 2026). PIRF likewise argues that naive reward backpropagation can lead to low sample efficiency, compromised data fidelity, and reward hacking, motivating layer-wise truncation and weight-based regularization (Yuan et al., 24 Sep 2025). These results indicate that stronger constraint formulations are not automatically better; the interaction between denoising objectives and physical penalties is itself an optimization problem.

A second misconception is that all physics-informed diffusion models use the same notion of “physics.” FunDiff emphasizes architectural priors and continuous function generation (Wang et al., 9 Jun 2025), PIDM and PILD center training-time residuals (Bastek et al., 2024, Zeng et al., 29 Jan 2026), PI-DDPM and PSDM enforce forward-model consistency during inference (Li et al., 2023, Han et al., 2024), while the SAR wake model transfers physics through simulator-generated data and parameter-derived prompts rather than explicit residual minimization (Kamirul et al., 28 Apr 2025). This suggests that the label is best understood operationally: what matters is where and how physical knowledge constrains the stochastic generative process.

The literature also reports clear limitations. FunDiff identifies handling complex geometries and extending to multiphysics as priority directions (Wang et al., 9 Jun 2025). PIDM notes grid-centric architectures and the need for residual evaluation to align with the data-generation mechanism (Bastek et al., 2024). The infrared PID model depends on the accuracy of the frozen TeV decomposition network and warns that excessive TeV-loss weighting can limit diversity (Mao et al., 2024). Pi-DPM for anomaly detection notes that anomaly injection is synthetic because genuine labeled spoofing data are rare, and that the physics prior is domain-dependent (Sharma et al., 8 Jun 2025). The SAR wake study reports blur and fine-scale deviations relative to the physics-based simulator, attributing them to modest dataset size and limited training duration (Kamirul et al., 28 Apr 2025).

The resulting research agenda is technically coherent. It points toward methods that preserve function regularity under diffusion, better balance data fidelity against constraint satisfaction, support arbitrary geometry and coupled physics, and reduce the approximation errors associated with intermediate value functions or late-stage residual evaluation. The field’s recent movement toward reward-based optimization, function-space diffusion, and spectral latent representations suggests a broader shift from post hoc physical correction toward generative models whose stochastic trajectories are physically meaningful throughout the denoising process (Yuan et al., 24 Sep 2025, Gallon et al., 10 Feb 2026).