partial differential equations

This work aims to uncover the unifying mathematical principles underlying several prominent approaches in generative modeling. By constructing a high-level framework grounded in optimal transport theory, it systematically elucidates the intrinsic connections among diffusion models, flow matching, and Schrödinger bridges. The study demonstrates that these seemingly distinct methods can all be interpreted as special cases of optimal transport problems under varying constraints or approximations. This unified perspective not only clarifies the fundamental commonalities shared by diverse generative models but also establishes a theoretical foundation for their integration and further innovation.

To address the performance bottleneck in optical quality control and 3D concrete crack segmentation caused by scarce annotated data, this paper proposes a physics-informed, data-driven multiscale unification framework. The method introduces a differentiable scale-bridging operator integrated with a conservation-law-preserving structured discretization paradigm, establishing—for the first time—a rigorous bidirectional equivalence between continuum models and discrete dynamical systems. This resolves critical challenges including constitutive closure deficiency and instability during scale transition. The framework synthesizes variational asymptotic analysis, Lie-group-based discrete mechanics, and topological dynamical systems theory. Evaluated on benchmark tasks—elastic wave propagation and lattice defect evolution—the approach achieves a 40% improvement in prediction accuracy and a 58% reduction in computational cost, while strictly enforcing energy and momentum conservation.

This work addresses the challenge of constructing effective mesoscopic dynamical equations for complex multiscale systems by proposing a hypothesis-driven modeling paradigm grounded in the generalized Onsager principle. Within a class of hypotheses that satisfy prior theoretical constraints—such as global well-posedness and asymptotic stability—the framework unifies the description of dissipative and conservative processes, integrating energy dissipation structure analysis with data-driven techniques to identify concrete models. Validated on both continuous PDE benchmarks and microscopic chain model data, the approach not only accurately reconstructs unknown mesoscopic dynamics but also yields physically interpretable diagnostic insights, thereby achieving a balanced trade-off among accuracy, robustness, and interpretability.

This work addresses the problem of efficiently and accurately bridging arbitrary probability density functions within a bounded time horizon. Methodologically, it introduces a unified generative modeling paradigm based on stochastic interpolation processes, seamlessly integrating flow-based and diffusion-based models—supporting both deterministic ordinary differential equation (ODE) paths and stochastic differential equation (SDE) paths with tunable noise. A novel score-matching objective is derived for the first time; theoretical analysis proves that optimizing only a quadratic loss suffices for likelihood control, overcoming the traditional limitation of deterministic models requiring additional Fisher divergence regularization. By unifying Schrödinger bridge theory, the Fokker–Planck equation, and variational inference, the framework rigorously recovers the Schrödinger bridge solution under optimal interpolation and provides a unified estimator for both likelihood and cross-entropy.

Data-driven discovery of partial differential equations (PDEs) remains challenging due to structural ambiguity and sensitivity to noise and data scale. Method: We propose an adjoint-based parametric modeling framework for PDE discovery. A sparse candidate library—comprising linear/nonlinear terms and spatial derivatives—is used to parameterize the PDE form, yielding a PDE-constrained optimization problem. For the first time, we systematically derive the adjoint equations for general parametric PDE families via variational calculus, enabling machine-precision analytical gradient computation. Contribution/Results: Our method significantly outperforms sparse regression approaches (e.g., PDE-FIND) in structural identification accuracy and noise robustness across diverse PDEs—including Burgers, KdV, and reaction-diffusion equations—especially under high noise levels and large-scale data. Integrated forward and adjoint numerical solvers ensure efficient training, while analytical gradients substantially accelerate optimization convergence.

This work investigates how to efficiently transform a simple prior distribution into a complex target distribution that satisfies boundary constraints via stochastic trajectories in probability space, while ensuring path optimality. Building upon Schrödinger bridge theory, we develop a first-principles generative modeling framework that achieves distributional transformation by minimizing entropy deviation. Our approach establishes a unified mathematical foundation linking Schrödinger bridges with modern generative models—including diffusion models, score matching, and flow matching—and introduces a generalizable, task-oriented dynamic construction method. By integrating optimal transport, stochastic control, and path-space optimization, we devise an efficient computational toolkit for dynamic Schrödinger bridges, offering both theoretical grounding and practical improvement pathways for existing generative models.

This work addresses the challenge of learning stochastic multiscale systems with unobserved fast processes from a single trajectory of slow variables, where the invariant distribution of the fast dynamics is unknown. The authors propose an end-to-end learning framework based on stochastic differential equations that integrates stochastic averaging for structure-preserving dimensionality reduction. Crucially, they introduce normalizing flows to flexibly parameterize the invariant distribution of the latent fast variables—a first in this context—and combine this with variational Bayesian inference to quantify epistemic uncertainty in model parameters. Using only a single observed slow-variable trajectory, the method accurately identifies the effective stochastic dynamics, achieving both scalability and significantly enhanced modeling robustness and reliability of uncertainty quantification.

This work investigates the theoretical foundation and thermodynamic optimality of the reweighting step in Population Annealing from the perspectives of optimal transport and nonequilibrium thermodynamics. By recasting the algorithm as a discrete-time Schrödinger bridge problem, we analytically solve the associated Schrödinger system and reveal that the reweighting mechanism arises from an optimal control potential on path space, naturally embedding thermodynamic work into a variational framework. For the first time, Schrödinger bridge theory is integrated with Population Annealing, unifying the geometry of optimal transport and nonequilibrium statistical mechanics, and clarifying the role of the Jarzynski equality within the Donsker–Varadhan variational principle. This analysis not only elucidates the origin of reweighting but also establishes its correspondence to a globally optimal control, thereby proving the algorithm’s theoretical optimality in sampling and optimization.

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.

This study addresses optimal control problems governed by semilinear partial differential equations (PDEs), presenting the first systematic comparison between direct and indirect physics-informed neural network (PINN) approaches. The direct method minimizes the objective functional while enforcing the state equation as a hard constraint, whereas the indirect method leverages Pontryagin-type first-order optimality conditions to jointly solve the state and adjoint equations. Using the Allen–Cahn equation as a benchmark, numerical experiments demonstrate that the indirect PINN formulation yields more accurate approximations of the true solution, produces smoother controls, and adheres more rigorously to both the PDE constraints and the underlying optimality structure. The work further reveals an implicit regularization effect inherent in PINN parameterization, underscoring the advantages of indirect methods in preserving physical consistency and optimality conditions.