PhysGen: Integrating Physics with Generative Modeling

• PhysGen is a framework combining data-driven generative models with explicit physics laws to simulate realistic physical phenomena.
• It merges techniques like GANs, diffusion models, and neural operators with physics-informed loss functions and hybrid grey-box methods.
• PhysGen enables faster, physically plausible simulations in applications such as image-to-video synthesis, surrogate modeling, and dynamics generation.

PhysGen refers broadly to methodologies, frameworks, and algorithms designed for generating, simulating, or modeling physical phenomena that explicitly incorporate both physical laws and data-driven learning. It aims to bridge the gap between data-centric generative modeling (such as image synthesis or surrogate modeling in physics) and the enforcement or extraction of underlying physical principles, such as those governed by ODEs, PDEs, or stochastic processes. Research under the umbrella of PhysGen spans rigid-body dynamics, continuous fields, spatiotemporal evolution, and multi-material systems, and employs tools ranging from classical numerical solvers and symbolic regression to modern neural operators, generative models, and hybrid (grey-box) approaches.

1. Definitions and Scope

PhysGen encompasses any generative approach to physics simulation, model discovery, or dynamics generation that combines data-driven components (e.g., neural networks, generative diffusion models, or symbolic regression) with an explicit integration of physics, either through physical residuals in the loss, modular physics encoding, or hybrid architectures. The objectives can include:

• Producing photorealistic or physically plausible videos/motions from images or 3D assets.
• Learning or inferring governing equations or hidden physics from data (sparse or dense).
• Accelerating simulations by replacing or augmenting traditional solvers with learned surrogates constrained by physics.
• Enabling interactive editing or real-time dynamics under user or environmental control.

2. Key Methodological Paradigms

Physics-Grounded Generative Modeling

PhysGen systems increasingly exploit generative models, such as GANs, diffusion models, and flow matching, with carefully constructed loss functions or architectures that encode either the governing equations (e.g., PDEs) or domain-specific constraints (Baldan et al., 10 Jun 2025). Frameworks such as Physics-Based Flow Matching (PBFM) explicitly add physics residuals to the generative objective and employ temporal unrolling during training to produce physically consistent samples.

Hybrid (Grey-Box) and Physics-Informed Models

Hybrid PhysGen frameworks address the issue of incomplete physics—where the analytical or computational model does not fully capture the underlying real-world behavior—by integrating a theory-driven (grey-box) module and a data-driven corrector, with distributions mapped via optimal transport (OT) to minimize distortion and preserve correct dependence on physics parameters (Singh et al., 27 Jun 2025). Such models excel in reconciling discrepancies between imperfect simulations and empirical data, yielding interpretable and physically valid generative outputs.

Symbolic Regression and Feature Engineering

Symbolic regression—employing methods such as Gene Expression Programming (GEP) and Sequential Threshold Ridge Regression (STRidge)—enables the automatic inference of explicit analytical forms for hidden physics or closure models from sparse or noisy measurements, with feature engineering playing a crucial role in both the completeness and interpretability of the discovered models (Vaddireddy et al., 2019).

Physics-Encoded Neural Operators

A major trend in PhysGen is the development of neural operators, such as graph neural operators (GNOs), that can model the solution operators of PDEs across varying geometries and boundary conditions. Physics- and geometry-aware operator architectures incorporate PDE residuals in physics-informed losses and utilize geometry-aware feature enrichment (e.g., interpolation-based or learnable encoders), enabling high data efficiency and generalization (Sarkar et al., 13 Aug 2025).

3. Core Model Structures and Training Approaches

Category	Core Model Structure	Physics Integration
Generative Surrogates	Diffusion, Flow Matching, U-Nets, GANs	Physics residual loss, temporal unrolling, stochastic sampling
Hybrid Grey-Box	fp (known physics) + fo (NN corrector)	OT-based mapping, explicit parameter conditioning
Symbolic Regression	GEP, STRidge, Expression Trees	Feature library reflecting physical derivatives/interactions
Neural Operators	Sp²GNO, πG-Sp²GNO, Graph Message Passing	Physics-informed/hybrid losses, geometry-encoded node features

During training, losses often combine standard distributional objectives (e.g., data likelihood or adversarial/OT losses) with physics-based residual terms, such as the deviation from satisfying the governing PDE at either collocation points or over a trajectory. Some methods, such as conflict-free gradient updates (Baldan et al., 10 Jun 2025), optimize both loss components in a way that avoids the need for manual balancing.

Temporal unrolling and higher-order time integration schemes further enhance the physical accuracy of predictions in time-dependent settings, while stochastic projection algorithms facilitate robust computation of spatial (and temporal) derivatives when activation functions prohibit automatic differentiation.

4. Applications of PhysGen

PhysGen frameworks support a wide range of applications:

• Physics-Grounded Image-to-Video and 3D-to-4D Synthesis: Systems such as PhysGen (Liu et al., 27 Sep 2024), PhysGen3D (Chen et al., 26 Mar 2025), and PhysGM (Lv et al., 19 Aug 2025) create temporally consistent and physically plausible animations from images or single-view 3D reconstructions. These methods employ rigid-body simulation, MPM, or constitutive Gaussian representations coupled with feed-forward or inference-time optimization strategies.
• Surrogate Modeling and Accelerated Simulation: PBFM (Baldan et al., 10 Jun 2025) and similar frameworks replace expensive numerical solvers for PDEs with fast, physically constrained surrogate generative models, enabling uncertainty quantification and efficient inverse design.
• Discovery of Physical Laws and Model Closure: Feature-engineering and symbolic regression techniques (Vaddireddy et al., 2019) extract interpretable constituent equations, closure terms, or hidden source terms given incomplete or noisy dynamics data.
• Generalization to Non-Standard Geometries or Sparse Data: Physics-aware neural operators (Sarkar et al., 13 Aug 2025) and message passing models (Zeng et al., 2 Oct 2024) enable accurate simulation on complex, irregular domains where classical mesh-based solvers or black-box neural models underperform.

5. Technical Innovations and Representative Formulations

Representative Equations and Algorithmic Strategies

• Flow Matching with Physics Residual:

$$L = w_{FM} \| u_t^\theta(x_t, t) - u_t(x_t) \|_2 + w_R \| \mathcal{R}(x_1(x_t, t)) \|_2$$

where $\mathcal{R}(\cdot)$ encodes the deviation from a PDE or algebraic constraint (Baldan et al., 10 Jun 2025).

• Optimal Transport Cost in Grey-Box Models:

$C_k(y, x) = k(y, y) + k(x, x) - 2k(y, x)$

and conditional distribution alignment as

$$\int p_o(y|x, \theta, z) p(x|x_0, \theta) p(\theta)p(x_0)p(z) dz dx d\theta \approx v(y)$$

(Singh et al., 27 Jun 2025).

• Stochastic Projection for Derivative Approximation:

$$\widehat{G}(\bar{x}) = \frac{ \frac{1}{N_b} \sum_i [u(x_i, \theta) - u(\bar{x}, \theta)](x_i - \bar{x})^T }{ \frac{1}{N_b} \sum_i (x_i - \bar{x})(x_i - \bar{x})^T }$$

(Sarkar et al., 13 Aug 2025).

Geometry-, Material-, and Multi-Physics Integration

PhysGen architectures increasingly address complex scenarios where:

• Geometry varies at inference (handled by geometry-aware encoding, e.g., πG-Sp²GNO),
• Material heterogeneity exists (handled by multi-material segmentation, expert model assignment, and learned weighting factors as in OmniPhysGS (Lin et al., 31 Jan 2025), Phys4DGen (Lin et al., 25 Nov 2024), PhysGM (Lv et al., 19 Aug 2025)),
• User-controllable simulation is needed (direct editing of initial speeds, material strengths, or external forces in PhysGen3D (Chen et al., 26 Mar 2025)).

6. Experimental Validation and Limitations

Across methods, PhysGen frameworks have demonstrated (when benchmarked against both classical and deep learning alternatives):

• Up to $8\times$ reduction in physical residuals compared to standard flow matching or diffusion models (Baldan et al., 10 Jun 2025).
• Robust generalization to unseen geometries and boundary conditions with state-of-the-art error reduction (Sarkar et al., 13 Aug 2025).
• Competitive or superior qualitative realism, text alignment, and user-preferred results relative to existing generative baselines in 4D dynamics, with generation speedups in the range of $10\times$–$100\times$ due to efficient feed-forward inference (Lv et al., 19 Aug 2025).
• Remaining limitations center on physical correctness for higher-order systems, effective scaling to highly multi-scale phenomena, and managing the trade-off between speed and strict law enforcement (Spitznagel et al., 7 Mar 2025).

7. Future Directions and Implications

The trajectory of PhysGen research supports increasing integration of:

• Hybrid surrogates combining physics laws with adaptive, data-driven corrections (grey-box OT frameworks),
• Automated, interpretable discovery of governing equations in complex systems through symbolic regression,
• Unified architectures for multi-physics, multi-material, and multi-scale environments,
• Foundation models for scientific computing where structure, uncertain data, dynamic parameterization, and domain knowledge are all simultaneously leveraged.

A plausible implication is that PhysGen frameworks will form the backbone of next-generation simulators, editors, and controllers in scientific, engineering, and creative domains where both the realism and controllability of generated physics are paramount.