transport equation analysis

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 simultaneously preserving conservation, entropy stability, and hyperbolicity in data-driven learning of hyperbolic conservation laws. To this end, the authors propose the SymCLaw framework, which parameterizes the flux function and jointly learns a convex entropy function along with its associated entropy potential. Without requiring prior knowledge of the governing equations, SymCLaw is the first data-driven approach to unify these three fundamental physical properties and automatically select physically admissible weak solutions. By integrating entropy-stable numerical fluxes with standard discretization-compatible techniques, the method demonstrates strong generalization to unseen initial conditions, robustness to noise, and high-accuracy long-time predictions across benchmark problems including Burgers’, shallow water, Euler, and KPP equations.

Traditional diffusion simulations rely on simplified constitutive models fitted to experimental data, often inducing information loss and model bias. This work proposes a data-driven mixed finite element framework that directly incorporates raw experimental measurements—bypassing parametric material modeling—and rigorously enforces conservation laws in strong form, thereby eliminating constitutive fitting. The method employs a mixed variational formulation to intrinsically guarantee conservation, relaxes regularity requirements on the solution space, and enables adaptive hp-refinement. Additionally, a posteriori error estimation quantifies solution non-uniqueness arising from data uncertainty. Validation on synthetic datasets and a nonlinear heat conduction problem in nuclear graphite demonstrates high-fidelity simulation accuracy, explicit characterization of multivalued solution structures under data-driven constraints, and faithful propagation of uncertainty. The framework establishes a new paradigm for experiment-guided modeling of diffusion processes.

This work systematically identifies and rectifies seven critical errors in Liu, Madhavan, and Tegmark’s machine learning–based discovery of conservation laws from a one-dimensional damped harmonic oscillator—including mis-specified physical priors, conflation of mathematical definitions of conserved quantities, inappropriate error metrics, and omission of differential equation validation. To address these, we introduce a physics-constrained error analysis framework, rigorous analytical verification, and formal scrutiny of conservation law definitions, thereby falsifying the original method’s applicability to dissipative systems. Our principal contributions are threefold: (i) establishing the first empirically testable theoretical validation standard for ML-driven conservation law discovery; (ii) rigorously distinguishing *invariance* (under symmetry transformations) from *conservation* (time-independence along trajectories); and (iii) proposing a robust modeling paradigm integrating differential geometry and dynamical systems theory—substantially enhancing methodological rigor and reproducibility in physics-guided machine learning.

This paper investigates the approximate controllability of the continuity equation under ReLU-type vector fields, aiming to construct a piecewise-constant time-varying control parameter θ = (w, a, b) that steers an initial density ρ_B arbitrarily close to a target density ρ_*. Using the relative entropy as the approximation error metric, the work establishes— for the first time—the approximability theory for ReLU vector fields in the relative entropy sense: arbitrary-precision approximation is achievable when the base and target distributions satisfy a relative tail-decay condition. It further provides an explicit upper bound on the number of control switches (i.e., segments of the piecewise-constant parameter), and characterizes the structure of the reachable set of the continuity equation under the relative entropy norm. This analysis furnishes a novel controllability-theoretic foundation and complexity guarantees for distribution transport via normalizing flows.

Traditional Eulerian approaches struggle to uncover the intrinsic transport mechanisms and coherent structures in unsteady heat conduction. This work proposes a Lagrangian visualization method based on massless particles, which tracks particle trajectories within a time-reparameterized spatiotemporal framework and accumulates their thermal transport contributions to identify attracting/repelling structures and coherent transport channels over finite time intervals. For the first time, this approach effectively extends the Lagrangian framework to non-periodic, non-conservative heat transport problems, thereby transcending the applicability limits of existing fluid mixing visualization techniques. The method successfully reveals dynamic thermal transport structures that are inaccessible to conventional approaches, significantly advancing the understanding of unsteady heat conduction mechanisms.

This work addresses the limitation of traditional lattice Boltzmann methods (LBM), which require manual construction of specialized schemes for each class of partial differential equations (PDEs), hindering efficient multiphysics modeling. The authors propose a unified framework that treats systems of conservation laws as discrete kinetic relaxation approximations, enabling automatic derivation of complete LBM formulations directly from conservative PDEs. By leveraging a direct flux-to-moment-space mapping and a convection-relaxation cascade mechanism, the approach intrinsically captures spatial gradients without ad hoc discretization. Built upon a coordinate-free domain-specific language (DSL), the symbolic compiler PDE2LBM automatically generates high-performance GPU kernels compatible with the OpenLB platform. Validation across twelve transport equations demonstrates second-order convergence in double precision, stable convergence in single precision, and GPU implementations achieving up to 96% of peak memory bandwidth.

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 the challenge of unified modeling of source/sink dynamics, cyclic behavior, and topology-constrained transport in complex dynamical systems. By integrating the continuous theory of Helmholtz–Hodge decomposition with discrete data-driven approaches, the authors propose a structured flow modeling paradigm based on graph vector fields (GVFs) and gradient–curl–harmonic decompositions over simplicial complexes. They develop a multi-level modeling strategy that spans from high expressivity to low computational cost through parameterized conditional models and a simplified Hodge representation. A cross-domain validation and diagnosis–simplification iterative pipeline is further designed to ensure model interpretability while achieving computational efficiency. The framework systematically elucidates the trade-offs among model complexity, interpretability, and predictive performance.

This work addresses the high computational cost of traditional numerical methods for solving parametric hyperbolic conservation laws and the tendency of existing neural surrogates to produce non-physical solutions, suffer from unstable rollouts, and exhibit poor generalization due to neglecting underlying physical structures. The authors propose a structure-preserving graph neural network (GNN) solver that explicitly formulates the GNN as a learnable conservative reconstruction and upwind flux operator, inherently satisfying local conservation and upwinding properties. By integrating this formulation within an ADER framework, the method enables high-order spatiotemporal predictions while maintaining physical consistency and supporting stable long-time integration with large time steps. On challenging supersonic flow benchmarks, the approach significantly outperforms both low-order numerical schemes and current neural surrogates in long-term prediction accuracy and achieves speedups of several orders of magnitude over conventional high-resolution simulations.