physics-informed generative modeling

This work addresses the challenge of simultaneously satisfying physical constraints and preserving data distribution fidelity in flow-matching generative models for scientific computing. Methodologically, it introduces a physics-informed post-training fine-tuning framework that embeds the weak-form residual of partial differential equations (PDEs) into a differentiable optimization objective—enhancing physical consistency without distorting the original distribution—and couples a learnable latent variable predictor to jointly optimize field solutions and unknown physical parameters (e.g., source terms, material properties, or boundary conditions). It is the first approach to unify flow-matching generative modeling with latent-parameter joint inverse inference within a single paradigm, enabling injection of physics priors without retraining. Evaluated on canonical PDE benchmarks, the method significantly improves PDE residual accuracy and parameter inversion fidelity, achieving both high-fidelity generation and robustness to ill-posed inverse problems—thereby advancing data-efficient, interpretable scientific discovery.

Pretrained flow-based generative models often fail to strictly satisfy nonlinear physical constraints—such as conservation laws or exact PDE solutions—due to inherent approximation errors. To address this, we propose Physics-Constrained Flow Matching (PCFM), a zero-shot framework that enforces hard physical constraints without model retraining or soft penalty terms. During continuous-time ODE sampling, PCFM dynamically injects exact constraint corrections via physics-driven intermediate-state projection and PDE-residual-guided gradient refinement. This is the first method to achieve zero-shot, uncompromising, and pointwise exact satisfaction of arbitrary nonlinear physical constraints throughout the entire generation process. Evaluated on diverse PDE modeling tasks featuring shocks, discontinuities, and sharp gradients, PCFM consistently outperforms unconstrained and soft-constrained baselines: it guarantees 100% hard-constraint compliance in final solutions while simultaneously improving sampling fidelity and physical consistency.

In scientific computing, generative models must strictly satisfy hard physical constraints—such as partial differential equations (PDEs)—yet gradient-based optimization is often infeasible due to gradient sparsity or high computational cost. Method: This paper proposes the Extrapolation–Correction–Interpolation (ECI) sampling framework, the first method to inject exact hard constraints into flow matching models *without gradients*, *without fine-tuning*, and *zero-shot*. Leveraging a pre-trained flow matching model, ECI iteratively refines generation trajectories via constraint-driven extrapolation, projection-based correction, and path interpolation—enforcing diverse PDE constraints rigorously while avoiding gradient computation and model adaptation. Contribution/Results: Experiments demonstrate that ECI achieves superior zero-shot generation quality over existing constrained generation baselines. Moreover, on regression tasks, it attains competitive accuracy without fine-tuning. ECI establishes an efficient, general-purpose paradigm for physics-informed generative modeling.

This work addresses the lack of physical plausibility in existing Transformer-based video generation models, which often disregard rigid-body physics during pixel-level denoising, leading to unrealistic behaviors in collision scenarios. To overcome this limitation, we propose a physics-aware reinforcement learning paradigm that, for the first time, explicitly embeds Newtonian mechanics–driven collision rules as reinforcement signals directly into the high-dimensional generative space, rather than imposing them as post-hoc constraints. We introduce the Mimicry-Discovery Cycle (MDcycle), a unified framework that preserves physical feedback during large-scale fine-tuning, enabling co-optimization of physical fidelity and generative flexibility. Experiments on the newly established PhysRVGBench benchmark demonstrate that our approach significantly outperforms current methods in both physical realism and rigid-body motion consistency.

This work addresses the lack of a unified understanding of the design principles, applicability, and performance differences between Physics-Informed Neural Networks (PINNs) and Neural Operators (NOs), which hinders the development of reliable data-driven PDE solvers. It proposes the first unified analytical framework that systematically characterizes the design space of both approaches along three dimensions: learning objectives, mechanisms for embedding physical structure, and strategies for computational load distribution. By elucidating the intrinsic connections and fundamental distinctions between these methods, the study not only clarifies the positioning and performance origins of existing techniques but also provides theoretical guidance and novel pathways for designing efficient and robust PDE solvers that effectively integrate physical priors with data-driven learning.

This work addresses the challenge of inverse parameter calibration in computational physics, where existing surrogate models often lack end-to-end differentiability and physical awareness, limiting their effectiveness. To overcome this, the authors propose a physics-informed latent-space framework based on an autoencoder architecture. The approach enables offline training of a differentiable surrogate model under observable supervision, mapping physical parameters to flow field predictions while embedding variational calibration directly in the latent space. By seamlessly integrating physical constraints with data-driven learning, the method achieves fully end-to-end differentiable surrogate modeling—a first in this domain. Evaluations on two computational fluid dynamics benchmarks demonstrate that, under realistic conditions including noise, low resolution, and partial observability, the proposed framework significantly reduces both calibration error and solution variability.

This work investigates whether pretrained image editing models can serve as a universal interface for solving diverse physical equations. The approach encodes both inputs and solutions of physical problems as images, incorporates lightweight adapters to embed scalar parameters, and trains the model under a unified architecture using numerical or analytical solutions across multiple equation types—including elliptic, heat, and Navier-Stokes equations. For the first time, it systematically demonstrates that general-purpose generative models can effectively represent both static and dynamic physical mappings, even capturing shocks and unstable phenomena, thereby expanding their applicability in scientific computing. Experiments across more than ten problem classes yield promising results, yet also reveal limitations of image-based representations in handling wide numerical ranges, enforcing constraints, and simulating long-term chaotic dynamics, such as those in the Kuramoto–Sivashinsky equation.

Traditional PDE solvers are computationally expensive, while existing learning-based solvers struggle with optimization in stiff, multiscale, or large-domain problems and fail to adequately capture uncertainty propagation. This work proposes “flow learners,” which directly model the continuous evolution between physically admissible states by parameterizing transport vector fields and integrating them to generate trajectories. Grounded in optimal transport theory, the approach seamlessly integrates physical constraints with generative modeling, yielding significant improvements over current learning-based solvers in continuous-time prediction, native uncertainty quantification, and long-term simulation of complex dynamics. The method establishes a new pathway toward building physics-aware PDE solution frameworks.

Existing surrogate models struggle to preserve essential physical consistency—such as conservation laws, invariants, and dissipative structures—in time-varying physical systems. This work proposes a latent-space surrogate operator framework that jointly trains an encoder, decoder, and latent flow map to explicitly constrain dynamics in the latent space, thereby accurately preserving or dissipating prescribed physical structures. The approach innovatively introduces a constrained transfer perspective, establishing a correspondence between physical structures in the original and latent spaces, and derives algebraic conditions for latent flow dynamics that guarantee preservation of linear or quadratic invariants or satisfaction of dissipation inequalities. Built upon a Latent Twin architecture with embedded physical constraints and structure-preserving optimization, the method significantly enhances physical consistency, structural fidelity, and long-term simulation stability on canonical ODE/PDE benchmarks while maintaining high predictive accuracy.

This work addresses the limited data efficiency and poor out-of-distribution (OOD) generalization of existing neural operator methods, which often neglect underlying physical principles—particularly when facing parameter variations or simulation-to-reality transfer. To overcome these limitations, we propose a multi-physics joint training framework that explicitly integrates the original partial differential equations (PDEs) with their simplified canonical forms directly into the neural operator training process. This architecture-agnostic approach is compatible with diverse neural operator designs and consistently enhances model robustness under parameter shifts and cross-domain scenarios. Extensive experiments across multiple 1D, 2D, and 3D PDE tasks demonstrate significant reductions in normalized root mean square error (nRMSE), confirming improved data efficiency and superior OOD generalization performance.