GenPhys: Generative Models with Physical Laws

• GenPhys is a framework where generative models are defined as the evolution of data under physical laws expressed through PDEs and conservation principles.
• It employs architectures like spectral generative flows and latent diffusion models that enforce physical constraints, ensuring stability and coherence in outputs.
• Physics-informed training and sampling techniques such as score matching and stochastic interpolants lead to improved uncertainty quantification and computational efficiency.

Generative Models from Physical Processes (GenPhys) refer to a paradigm in which the structure and training of generative models are fundamentally informed or dictated by underlying physical laws—typically in the form of partial differential equations (PDEs), conservation constraints, or physically meaningful field dynamics. GenPhys frameworks generalize and unify families such as score-based diffusion models, Poisson flow models, stochastic interpolants, and recent spectral generative architectures, providing rigorous mechanisms for enforcing physical consistency, inductive bias, and stability in generative modeling.

1. Foundational Principles and Formalism

The core premise of GenPhys is to recast data generation as the evolution of a field or configuration under a physical process—often encoded as a stochastic, dissipative, or otherwise smoothing PDE. In a widely cited generalization, a PDE admitting a “density-flow” form,

$$\frac{\partial p}{\partial t} + \nabla\cdot(p v) - R = p_{\text{data}}(x) \delta(t),$$

with $p$ a density, $v$ a velocity field, and $R$ a birth/death or source term, can serve as the forward “noising” process. Under suitable smoothing criteria—specifically, that high-frequency Fourier modes decay faster than low modes—such an s-generative PDE guarantees convergence to a simple prior at large times and an invertible generative flow back to the data distribution. This construction underpins not just classical diffusion models based on the heat equation but also more general flows derived from electrostatics (e.g., Poisson and Yukawa equations), and stochastic dynamics in physically meaningful spaces (Liu et al., 2023).

Backward sampling may proceed via deterministic ODEs (probability-flow) or SDEs (score-based reversal), with neural networks trained to either approximate the physical velocity field $v(x,t)$ (“flow matching”, “field matching”) or the score function $\nabla_x \log p(x,t)$ (“score matching”) (Hein et al., 2024). In all cases, the generative transformation is tightly connected to the physics of the underlying field, such as conservation, dissipation, or multiscale structure.

2. Model Architectures and Physical Representations

GenPhys encompasses a variety of architectures, each reflecting different levels of physical abstraction:

• Field-Theoretic and Wavelet Representations: Spectral Generative Flow Models (SGFMs) treat text or video as the evolution of continuous fields governed by constrained stochastic PDEs, with text/video unified as trajectories in function space. Multiresolution wavelet bases are used for efficient representation and scale separation, allowing deterministic generation of coarse (global semantic) modes and stochastic refinement of local detail (Kiruluta, 13 Jan 2026).
• Latent-Function Spaces: FunDiff extends score-based generative models to the infinite-dimensional setting by combining latent diffusion processes with function-space autoencoders (FAEs) capable of resolution-invariant, continuous query. Physical priors are embedded directly into the encoder/decoder or loss via analytic constraints (e.g., divergence-free velocity in fluid dynamics, symmetry, periodicity) or soft PDE residual penalties (Wang et al., 9 Jun 2025).
• Operator and Surrogate Models: GenPhys frameworks also encompass semi-supervised surrogates for coarse-grained physics, combining generative latent variables with explicit physics solvers (e.g., via “virtual observables” as additional likelihood terms), and probabilistic integration between data-driven and physically-derived outputs (Rixner et al., 2020, Spitieris et al., 25 May 2025).
• Localized Out-of-Equilibrium Protocols: Physical generative models can be realized in analog or hardware platforms as networks of coupled nonlinear oscillators or more general stochastic dynamical systems, with local learning rules operating directly on measured or simulated forces/energies (Bösch et al., 23 Jun 2025).

3. Physics-Informed Training and Sampling Procedures

GenPhys models incorporate physical structure through training objectives, architecture, and sampling:

• Physics-Enforced Losses and Constraints: Typical schemes augment data likelihood or evidence lower bound (ELBO) objectives with regularization enforcing physical residuals, as in physics-informed score-based models (Yang et al., 2018), PG-GANs with physics-guided discriminators (Yonekura, 2023), and constrained stochastic flows in wavelet space (Kiruluta, 13 Jan 2026). Hard constraints (e.g., divergence-freeness, exact symmetry) can be embedded in decoder architectures.
• Stochastic Interpolants: Recently, direct stochastic interpolants between temporally or physically adjacent states have been shown to dramatically improve forecasting and generative emulation of physical systems. Rather than denoising from a distant isotropic prior, interpolants exploit the physical proximity of successive PDE states, with learnable drift fields or energy-consistent parameter schedules imposing constraints such as energy conservation and divergence-free evolution (Zhou et al., 30 Sep 2025, Mücke et al., 8 Apr 2025).
• Physics Consistency at Inference: Some frameworks, such as CoCoGen, apply physically-consistent residual minimization—by subtracting the gradient of the PDE residual norm at every sampling step—guaranteeing final outputs consistent with governing equations to prescribed tolerance (Jacobsen et al., 2023).
• Multi-Fidelity and Bayesian Enhancements: Diffusion-based surrogates for multi-fidelity systems incorporate inexpensive physical simulations as cheap context and refine via expensive solver data through guided score updates. These approaches admit theoretical Wasserstein error guarantees and enable rigorous uncertainty quantification (Shi et al., 2024).

4. Comparative Analysis and Empirical Findings

Comparisons across classes of GenPhys models consistently highlight several advantages:

• Long-Range Coherence and Stability: SGFMs and energy-consistent interpolant models demonstrate robust long-range coherence and bounded energy dynamics, in contrast to the drifting or mode collapse seen in autoregressive and naive diffusion architectures (Kiruluta, 13 Jan 2026, Mücke et al., 8 Apr 2025).
• Physical Consistency and Uncertainty: Embedding PDE constraints, conservation laws, or divergence-free projections leads to outputs with statistically and physically calibrated behavior—lower physical-residual RMSE, improved conservation error, and sharper sample distributions (Wang et al., 9 Jun 2025, Jacobsen et al., 2023, Zhou et al., 2024).
• Computational Efficiency: Leveraging spectral projection (e.g., via wavelet basis), latent diffusion in function space, and direct stochastic interpolants permits sub-quadratic scaling with context length or spatial resolution, outperforming standard transformer and pixel-based approaches (Kiruluta, 13 Jan 2026, Wang et al., 9 Jun 2025).
• Sample Diversity and Structure: GenPhys models routinely produce samples that not only match global statistics (MSE, likelihood) but preserve fine-scale physical structures (vorticity, coherent turbulence, contextual consistency), even after hundreds of rollout steps (Mücke et al., 8 Apr 2025, Zhou et al., 30 Sep 2025).
• Empirical Benchmarks: For Kolmogorov flow, energy-consistent SI models maintain spectral and field-wise errors below those of ACDM and PDE-Refiner baselines (LSiM error ≈ 0.05, kinetic-energy W₁ error ≈ 2%) (Mücke et al., 8 Apr 2025). For climate emulation and weather forecasting, stochastic interpolants achieve lower VRMSE and CRPS with fewer neural function evaluations than DDPM/EDM baselines (Zhou et al., 30 Sep 2025).

5. Domains of Application and Generalization

GenPhys methods are established in diverse physical contexts:

• Fluid Dynamics: Turbulent flow emulation with incompressibility, spectral energy, and stability constraints; rollouts of Kolmogorov and Boussinesq flows (Wang et al., 9 Jun 2025, Mücke et al., 8 Apr 2025, Zhou et al., 30 Sep 2025).
• Elastodynamics and Solid Mechanics: Variational-structured models for 4D elastodynamics embedding local equilibrium at every neural operation, generalizing to arbitrary material laws (Feng et al., 2024).
• Hydrology, Cosmology, and Astrophysics: Generative surrogates incorporating physical latent structure for rainfall-runoff sequence generation and for hypothesis testing in systems like galaxy evolution (Yang et al., 2023, Schawinski et al., 2018).
• Medical Imaging and Sensory Data: Diffusion and Poisson-flow generative models for Bayesian image reconstruction and data enhancement in imaging modalities (Hein et al., 2024).
• Inverse Problems and Surrogate Modeling: Physics-informed score models with flexible conditioning and uncertainty quantification enabling forward, inverse, and multi-fidelity surrogate tasks (Jacobsen et al., 2023, Shi et al., 2024).

Generalization across domains is facilitated by working in function space rather than discretized or pixelized representations, leveraging continuous-query architectures and domain-agnostic operator learning (Wang et al., 9 Jun 2025, Kiruluta, 13 Jan 2026).

6. Design Space, Limitations, and Future Directions

The theory and practice of GenPhys suggest a rapidly expanding design space:

• Expanding the PDE Zoo: Any s-generative PDE (strictly dissipative dispersion relation) yields a valid foundation for a generative model; new families include Yukawa flows and hybrid/fractional PDEs (Liu et al., 2023).
• Nonlinear and Coupled Physics: Open questions remain in extending GenPhys to nonlinear propagators (Navier–Stokes, reaction-diffusion), coupled multi-physics (thermo-electro, fluid-structure), and more complex boundary and source conditions (Wang et al., 9 Jun 2025).
• Structured Priors and Equivariant Architecture: Embedding continuous symmetry (e.g., $SE(n)$, permutation), hard divergence constraints, and physically meaningful coordinate embeddings (e.g., streamfunction, symmetry) is critical for further advances in both expressivity and inductive bias (Zhou et al., 2024, Wang et al., 9 Jun 2025).
• Computational Scalability and Hardware Realization: Function-space diffusion, spectral transforms, and local learning rules (even in analog oscillator networks) support scalable and potentially hardware-accelerated generative modeling of physical processes (Bösch et al., 23 Jun 2025).
• Uncertainty Quantification and Bayesian Analysis: GenPhys provides a rigorous route to uncertainty quantification in both forward and inverse settings, with explicit posterior sampling and Wasserstein error bounds (Hein et al., 2024, Shi et al., 2024).
• Limitations: Some models remain sensitive to autoencoder reconstruction error, architectural Lipschitz constants, cost of latent ODE sampling, and the tractability of large-scale PDE constraint enforcement. Trade-offs between sample fidelity, physical consistency, and computational cost are not fully resolved (Wang et al., 9 Jun 2025, Kiruluta, 13 Jan 2026).

GenPhys thus comprises a unified, physically grounded framework empowering rigorous generative modeling that is robust, physically interpretable, and computationally efficient, marking a substantive departure from conventional autoregressive and purely data-driven paradigms (Kiruluta, 13 Jan 2026, Liu et al., 2023, Wang et al., 9 Jun 2025, Mücke et al., 8 Apr 2025, Hein et al., 2024).