Physics-Conditioned Generative Models

Updated 8 July 2026

Physics-conditioned generative models are systems that condition outputs on physical variables, simulator data, and governing equations.
They employ diverse architectures like graph neural networks, conditional VAEs, and diffusion models to capture complex physical dynamics.
Empirical studies in particle physics, flood mapping, and video control show enhanced physical fidelity and improved constraint satisfaction.

Searching arXiv for papers on physics-conditioned generative models and related terminology. A physics-conditioned generative model is a generative model whose output distribution is conditioned on physically meaningful variables, simulator outputs, governing equations, or physically grounded annotations, so that sampling is tied to domain structure rather than to unconstrained data likelihood alone. In recent work, this notion has been instantiated for conditional detector simulation of unordered particle sets $p(R\mid T)$ , PDE-governed forward and inverse problems, flood mapping from topography and rainfall, controllable video generation from forces and material parameters, seismic waveform synthesis from arrival labels, galaxy synthesis from redshift, and biological sequence design from peptide–HLA context (Bello et al., 2022, Jacobsen et al., 2023, Wu et al., 12 Nov 2025, Wang et al., 24 Sep 2025, Chen et al., 2023, Li et al., 2024, Ma et al., 7 Oct 2025). The field therefore spans conditional VAEs, conditional Wasserstein GANs, score-based and diffusion models, flow matching, mixture density formulations, and grey-box optimal-transport schemes, with conditioning applied either during training, during sampling, or at both stages.

1. Problem classes and formalizations

Physics-conditioned generation appears in several distinct formal problem classes. In particle physics fast simulation, the task is to approximate the conditional distribution $p(R\mid T)$ , where both $T=\{t_i\}_{i=1}^{N_T}$ and $R=\{r_i\}_{i=1}^{N_R}$ are unordered sets of variable size. This formulation is motivated by detector response modeling at the Large Hadron Collider, where one wishes to emulate the conditional set-to-set mapping from truth particles to reconstructed particles without running the full high-dimensional simulation of detector hits (Bello et al., 2022).

In physics-based inverse problems, the objective is typically to sample from a posterior distribution $p(x\mid y)$ or $\mu_{X\mid Y}$ , where $x$ denotes an unknown field or parameter and $y$ a measurement generated by a forward operator constrained by physical principles. Conditional Wasserstein GANs have been used in this setting to learn posterior samplers directly from joint samples $(x,y)$ , replacing MCMC-style posterior sampling by fast conditional generation (Ray et al., 2022, Ray et al., 2023). This same conditional structure also appears in probabilistic field reconstruction and inversion from sparse measurements for PDE systems, where parameters, macroscopic quantities, or partial field observations serve as conditioning variables (Jacobsen et al., 2023).

A second major class comprises forward surrogate models. Here the condition is a physical specification of the system, and the output is a solution field, motion trajectory, or map. Examples include Darcy-flow fields conditioned on parameters or sparse observations (Jacobsen et al., 2023), flood depth maps conditioned on DEM, rainfall time series, and a simplified inundation model prior (Wu et al., 12 Nov 2025), physics-based character control conditioned on state-dependent latent skills (Yao et al., 2022), and earthquake ground motion waveforms conditioned on magnitude, depth, and source–station coordinates (Ren et al., 2024). In multimodal scientific systems, the conditional target may itself be intrinsically multimodal, and mixture density formulations have been proposed to model $p(u\mid x)$ explicitly as a mixture rather than forcing a unimodal approximation (Han et al., 11 Feb 2026).

A third class uses conditioning to express physical control rather than merely physical context. Video models have been conditioned on force vectors, material parameters, boundary conditions, and initial point clouds to generate physically grounded 3D motion trajectories (Wang et al., 24 Sep 2025). Related work conditions video diffusion on chunk-wise local physics descriptions or on explicit “goal force” tensors specifying desired effects on objects over spacetime (Saurabh et al., 27 Mar 2026, Gillman et al., 9 Jan 2026). In biological design, conditioning may combine sequence context with residue-level physicochemical descriptors, as in dual-conditioned T-cell receptor generation (Ma et al., 7 Oct 2025).

2. Conditioning interfaces

The conditioning signal may be symbolic, geometric, field-based, simulator-derived, or distributional. Across the literature, the interface between physical knowledge and generative modeling is highly heterogeneous.

Conditioning source	Representative use	Papers
Truth particle set $p(R\mid T)$ 0	Generate reconstructed particle set $p(R\mid T)$ 1	(Bello et al., 2022)
Parameters, macroscopic quantities, sparse field measurements	Forward surrogates and inverse PDE reconstruction	(Jacobsen et al., 2023)
DEM, rainfall time series, SPM output	Flood depth prediction	(Wu et al., 12 Nov 2025)
$p(R\mid T)$ 2	3D point-trajectory generation for video control	(Wang et al., 24 Sep 2025)
Redshift	Conditional galaxy image generation	(Li et al., 2024)
$p(R\mid T)$ 3	Labeled seismic waveform synthesis	(Chen et al., 2023)
Peptide and HLA context	TCR sequence generation	(Ma et al., 7 Oct 2025)

Set-valued conditioning requires permutation invariance or equivariance. The particle-physics model based on a graph neural network encoder and slot-attention generator makes the input truth particles available both through a global event embedding $p(R\mid T)$ 4 and through slot attention over particle representations, while preserving the fact that both input and output are unordered sets (Bello et al., 2022). This design is distinct from conventional vector-valued conditioning because the conditioning object is itself a variable-cardinality structured set.

Field-valued conditioning is common in PDE and environmental models. CoCoGen conditions score-based models on parameters, macroscopic quantities, or sparse measurements, while PIFF conditions a conditional flow-matching model on the current state $p(R\mid T)$ 5, the DEM $p(R\mid T)$ 6, time $p(R\mid T)$ 7, rainfall history $p(R\mid T)$ 8, and the SPM flood prior $p(R\mid T)$ 9 through the vector field $T=\{t_i\}_{i=1}^{N_T}$ 0 (Jacobsen et al., 2023, Wu et al., 12 Nov 2025). In multi-fidelity calibration, low-fidelity simulator outputs are injected directly into the denoising network, while high-fidelity simulator outputs act as guidance at inference rather than as ordinary conditioning inputs (Shi et al., 2024).

Textual and annotation-based conditioning can also be made physics-specific. PhysVid divides videos into temporally contiguous chunks, annotates each chunk with descriptions of dynamics, shape, and optics, and aligns these local prompts with corresponding video chunks through chunk-aware cross-attention (Saurabh et al., 27 Mar 2026). Goal Force instead uses a three-channel physics control tensor $T=\{t_i\}_{i=1}^{N_T}$ 1 whose channels encode direct force, goal force, and mass (Gillman et al., 9 Jan 2026). These approaches suggest that conditioning can target not only a final physical state but also intermediate local interactions or intended causal effects.

3. Mechanisms for embedding physics

Physics conditioning does not imply a single enforcement mechanism. The literature separates at least four broad modes: implicit conditioning through physics-generated data, explicit training-time penalties, adversarial or external judging, and sampling-time correction or projection.

Implicit conditioning uses a physically generated joint dataset without inserting equations directly into the loss. In the cWGAN approach to physics-based inverse problems, the physics constraint enters through the data generation process: all training pairs are generated using the true forward map $T=\{t_i\}_{i=1}^{N_T}$ 2, and no additional explicit physics term is added to the loss (Ray et al., 2023). CGM-GM likewise aims to learn complex wave physics and Earth heterogeneities from conditioning on magnitude and geographic coordinates, explicitly stating that this is done “without explicit physics constraints” (Ren et al., 2024). This mode treats physics as structure in the conditional data distribution.

Explicit training-time enforcement inserts residuals or physically grounded penalties into the objective. PhysCtrl adds a velocity loss, a Material Point Method-based physics loss on deformation gradients, and a boundary loss preventing penetration through the floor (Wang et al., 24 Sep 2025). The multimodal conditional mixture model introduces a distribution-level prior

$T=\{t_i\}_{i=1}^{N_T}$ 3

with a physics regularization term weighted by active component probabilities, thereby enforcing physical structure at the level of mixture means rather than individual samples (Han et al., 11 Feb 2026). The structure-preserving consistency framework for PDEs adds PDE residual penalties only in a second training stage and only on the active solution decoder, after freezing the coefficient decoder (Chang et al., 10 Feb 2026).

External-judge approaches separate the physical model from the differentiable generator. PG-GAN uses physical equations outside the neural network to determine whether generated samples are physically reasonable, defining sets $T=\{t_i\}_{i=1}^{N_T}$ 4 and $T=\{t_i\}_{i=1}^{N_T}$ 5 according to a residual threshold $T=\{t_i\}_{i=1}^{N_T}$ 6, and training the discriminator to mimic that evaluation (Yonekura, 2023). This allows black-box or commercial solvers to provide supervision without being embedded in the computational graph.

Sampling-time enforcement modifies the generative trajectory after training. CoCoGen uses a physical consistency step during the final stages of reverse-time sampling,

$T=\{t_i\}_{i=1}^{N_T}$ 7

where $T=\{t_i\}_{i=1}^{N_T}$ 8 is the discrete PDE residual (Jacobsen et al., 2023). PCFM generalizes this idea to a zero-shot inference framework for pretrained flow-based models, using Gauss–Newton-style projection and a final hard projection so that arbitrary nonlinear constraints can be satisfied exactly at the final solution (Utkarsh et al., 4 Jun 2025). The diffusion-based surrogate for multi-fidelity calibration similarly guides denoising trajectories with high-fidelity simulator information only at inference time (Shi et al., 2024).

A recurrent misconception is that conditioning on physical inputs automatically guarantees physical correctness. Several papers directly reject that equivalence. PhysVid begins from the observation that high-fidelity generative video models often violate basic physical principles, despite standard conditioning (Saurabh et al., 27 Mar 2026). PCFM argues that soft penalties and architectural biases often fail to guarantee hard constraints (Utkarsh et al., 4 Jun 2025). CoCoGen, PG-GAN, and sCM-PINN each address the same gap from different directions: residual correction during sampling, external physical judging, and structure-preserving physics-informed fine-tuning (Jacobsen et al., 2023, Yonekura, 2023, Chang et al., 10 Feb 2026).

4. Architectural patterns

The architectural design of physics-conditioned generative models follows the structure of the conditioning signal and the enforcement mechanism.

For unordered structured data, graph-based and set-based encoders are prominent. The particle-physics set-conditional set generator uses a fully connected graph over truth particles, MLP-based node embeddings, message passing, permutation-invariant pooling to obtain a global event representation $T=\{t_i\}_{i=1}^{N_T}$ 9, and a slot-attention generator in which $R=\{r_i\}_{i=1}^{N_R}$ 0 slots are initialized with random noise and embedded indices (Bello et al., 2022). By contrast, the baseline cVAE in the same study uses Deep Sets for encoding $R=\{r_i\}_{i=1}^{N_R}$ 1 and $R=\{r_i\}_{i=1}^{N_R}$ 2, with a presence variable to mask padded outputs. The paper’s comparison is architecturally significant because it isolates the role of hierarchy and attention in conditional set generation.

For inverse problems on fields and images, conditional adversarial models and conditional VAEs remain common. The cWGAN of posterior inference in PDE-based inverse problems uses a U-Net generator with conditional instance normalization,

$R=\{r_i\}_{i=1}^{N_R}$ 3

which injects stochasticity at multiple scales and decouples latent dimension from measurement dimension (Ray et al., 2022). The related cWGAN with full gradient penalty instead emphasizes critic design: its critic is required to be 1-Lipschitz with respect to both inferred and measurement vectors, not just the former (Ray et al., 2023).

Score-based, diffusion, and flow-matching models dominate recent PDE and spatiotemporal work. CoCoGen builds unconditional score models and then trains a conditional augmentation, ControlNet-style, on top of a frozen unconditional model (Jacobsen et al., 2023). PIFF uses image-to-image conditional flow matching, linear interpolation between DEM and flood maps, a transformer encoder for the rainfall sequence, and stepwise conditioning on the SPM prior (Wu et al., 12 Nov 2025). PhysCtrl represents dynamics as 3D point trajectories over 2048-point clouds and uses a Diffusion Transformer with spatiotemporal attention blocks, where spatial attention models interactions among points within a frame and temporal attention aggregates a point’s state across timesteps (Wang et al., 24 Sep 2025).

Sequential latent-variable architectures remain important when the physical signal is fundamentally temporal. CGM-GM uses a dynamic VAE on STFT amplitude spectrograms, with GRU-based encoder and prior networks conditioned on earthquake metadata (Ren et al., 2024). PhaseGen uses a conditional GAN over three-component seismic traces, conditioned on $R=\{r_i\}_{i=1}^{N_R}$ 4 and trained with WGAN-GP despite using only 100 labeled seismic events (Chen et al., 2023). ControlVAE combines a state-conditional prior

$R=\{r_i\}_{i=1}^{N_R}$ 5

with a learnable world model $R=\{r_i\}_{i=1}^{N_R}$ 6, producing a model-based generative control policy rather than a static data generator (Yao et al., 2022).

Alternative formulations emphasize interpretability. The MDN framework parameterizes the full conditional law explicitly as a small mixture with componentwise physics regularization (Han et al., 11 Feb 2026). The hybrid grey-box OT model composes incomplete physics $R=\{r_i\}_{i=1}^{N_R}$ 7 with a learned completion $R=\{r_i\}_{i=1}^{N_R}$ 8, keeping physics parameters $R=\{r_i\}_{i=1}^{N_R}$ 9 as explicit inputs so that the learned map augments rather than overrides known dynamics (Singh et al., 27 Jun 2025). GenCP extends conditional modeling to coupled multiphysics by learning decoupled conditional velocity fields and recombining them during sampling through Lie–Trotter operator splitting in measure space (Gao et al., 27 Jan 2026).

5. Applications and reported performance

Empirical results show that physics conditioning is used both to improve fidelity under standard generative metrics and to improve specifically physical metrics that unconstrained models can satisfy poorly.

In particle-physics detector simulation, the GNN plus slot-attention model reports cardinality prediction accuracy of $p(x\mid y)$ 0 versus $p(x\mid y)$ 1 for the cVAE baseline, $p(x\mid y)$ 2 versus $p(x\mid y)$ 3, and Hungarian cost $p(x\mid y)$ 4 versus $p(x\mid y)$ 5. The qualitative conclusion is that both models reproduce marginal per-particle distributions, but only the GNN+SA model accurately reproduces conditional statistics and inter-particle correlations (Bello et al., 2022).

In flood-depth mapping, PIFF is reported to outperform both pure physics-based and non-physics-informed AI baselines on real-event rainfall data, with $p(x\mid y)$ 6, $p(x\mid y)$ 7, and $p(x\mid y)$ 8. By comparison, SPM alone is reported as $p(x\mid y)$ 9, $\mu_{X\mid Y}$ 0, and $\mu_{X\mid Y}$ 1. Reported generation time per image is approximately $\mu_{X\mid Y}$ 2 seconds for PIFF, $\mu_{X\mid Y}$ 3– $\mu_{X\mid Y}$ 4 seconds for SPM, and approximately $\mu_{X\mid Y}$ 5 seconds for TUFLOW (Wu et al., 12 Nov 2025).

In controllable video and motion generation, PhysCtrl reports trajectory-generation metrics of $\mu_{X\mid Y}$ 6, $\mu_{X\mid Y}$ 7, and $\mu_{X\mid Y}$ 8, compared with $\mu_{X\mid Y}$ 9, $x$ 0, and $x$ 1 for Motion2VecSets and $x$ 2, $x$ 3, and $x$ 4 for MDM. In GPT-4o evaluation, PhysCtrl receives 4.5 for Semantic Adherence, 4.5 for Physical Commonsense, and 4.3 for Video Quality; user study results report 81% selection for physics plausibility as “best” and 66% for video quality (Wang et al., 24 Sep 2025). PhysVid, using chunk-wise local physics prompts and negative physics prompts, reports physical commonsense scores of 0.32 on VideoPhy and 0.64 on VideoPhy2, representing approximately 33% and up to approximately 8% gains over baseline generators in those benchmarks (Saurabh et al., 27 Mar 2026).

In PDE-focused work, CoCoGen reports residuals comparable to data generated through conventional PDE solvers and states that physical consistency steps reduce the average residual in generated samples to match or surpass those of finite-difference-based PDE solutions (Jacobsen et al., 2023). PCFM reports exact final constraint satisfaction together with improved MMSE, SMSE, and FPD on heat, Navier–Stokes, reaction–diffusion, and Burgers benchmarks, and emphasizes that it is uniquely able to capture sharp shocks while satisfying multiple constraints exactly (Utkarsh et al., 4 Jun 2025). The structure-preserving consistency framework reports 42% lower relative $x$ 5 error than the data-only baseline and few-step inference with 2 versus 63 NFEs in comparison with diffusion baselines (Chang et al., 10 Feb 2026).

Seismological and astrophysical applications further show that evaluation often shifts from perceptual plausibility to domain-specific metrics. PhaseGen reports a mean maximum normalized cross-correlation of 0.84 over 1000 generated seismic samples and mean discrepancy of approximately 8 ms for P picks and approximately 30 ms for S picks relative to human annotation (Chen et al., 2023). In galaxy generation conditioned on redshift, the DDPM reports Galaxy KL Loss 0.22 versus 0.23 for the CVAE, Galaxy Fitting Loss 5.32 versus 10.2, and FID 98.8 versus 123, whereas both models perform poorly on Redshift Loss, 0.62 for DDPM and 0.59 for CVAE (Li et al., 2024). This comparison is notable because the paper explicitly argues that physics-based metrics can discern strengths and weaknesses that human evaluation alone does not capture.

6. Limitations, methodological tensions, and current directions

A central methodological tension concerns soft versus hard physics. PINN-style residual penalties, adversarial residual surrogates, and physics losses during training can improve average physical fidelity, yet several recent works state that these mechanisms do not ensure exact satisfaction of constraints (Yonekura, 2023, Utkarsh et al., 4 Jun 2025). This has led to a growing emphasis on post-hoc or sampling-time correction, including PDE residual descent, projected flow trajectories, and guided multi-fidelity denoising (Jacobsen et al., 2023, Utkarsh et al., 4 Jun 2025, Shi et al., 2024). A plausible implication is that, in applications where conservation laws or boundary conditions must hold exactly, inference-time correction is becoming as important as training-time conditioning.

A second tension concerns stability. The consistency-model framework identifies a failure mode in which PDE residuals can drive the model toward trivial or degenerate solutions, motivating a two-stage procedure that freezes coefficient channels during physics-informed fine-tuning (Chang et al., 10 Feb 2026). PG-GAN similarly requires pre-training and threshold scheduling because early training can stagnate if no generated sample initially falls within the physical tolerance set (Yonekura, 2023). PhaseGen reports that conditioning outside the range covered by the small training set leads to mode collapse or failed generation (Chen et al., 2023). These results indicate that physics conditioning may regularize generation, but it can also create brittle optimization landscapes.

A third tension concerns the relationship between explicit physics and learned structure. Some models aim to learn physics indirectly from richly conditioned data, as in CGM-GM’s use of coordinates, magnitudes, and depths without explicit constraints (Ren et al., 2024). Others insist on explicit physical compliance, as in CoCoGen, PCFM, and PhysCtrl (Jacobsen et al., 2023, Utkarsh et al., 4 Jun 2025, Wang et al., 24 Sep 2025). The literature therefore does not support a single definition of “physics-informed” or “physics-conditioned”; rather, it spans a continuum from physically indexed conditional density estimation to hard-constrained generative simulation.

Current directions include coupled multiphysics and incomplete-physics settings. GenCP proposes training on decoupled data and inferring coupled physics during sampling, with an operator-splitting error bound of the form

$x$ 6

thus making “conditional-to-joint” generation mathematically explicit (Gao et al., 27 Jan 2026). The hybrid grey-box OT model addresses unpaired data and missing physics terms by using OT maps in data space while preserving correct usage of physics parameters (Singh et al., 27 Jun 2025). These developments suggest a broadening of the topic from conditioning on known physical descriptors toward generative completion of partial, decoupled, or imperfect physics models.

Physics-conditioned generative modeling is therefore best understood not as a single algorithmic family but as a design principle: the conditional distribution is structured so that generation depends on variables, operators, and constraints that carry physical meaning. The concrete implementation may be a graph encoder, a U-Net with conditional instance normalization, a diffusion transformer with spatiotemporal attention, a frozen score model with conditional augmentation, a mixture density network with distribution-level priors, or a grey-box transport map. What unifies these systems is the attempt to align probabilistic generation with the symmetries, conservation laws, causal mechanisms, and latent regimes of physical domains.