arXiv subjects

Bingcheng Hu

Publications and source records attributed to Bingcheng Hu.

5 recordsLinked to original sources

arXiv2026

TensorSpectra: Bridging the Spectral and Tensor Worlds for High-Dimensional PDEs

High-dimensional partial differential equations (PDEs) remain challenging for spectral methods due to the exponential growth of multidimensional coefficient spaces. We propose TensorSpectra, a low-rank spectral framework that directly parameterizes spectral coefficient tensors in canonical polyadic (CP) form, reducing representation complexity from $\mathcal{O}(N^d)$ to $\mathcal{O}(dNr)$. TensorSpectra combines structure-preserving spectral calculus, de-aliased nonlinear interactions, operator-aware residual weighting, and progressive low-rank corrections to enable efficient PDEs solving in compressed coefficient space. We further establish a theoretical framework connecting spectral truncation, low-rank approximation, and optimization errors. Extensive experiments across five PDE families and 24 benchmark configurations demonstrate State-Of-The-Art accuracy among five representative baselines, with TensorSpectra consistently achieving the lowest relative $L^2$ errors. More broadly, our findings offer a new perspective on high-dimensional PDEs solving, suggesting that exploiting intrinsic low-rank structure in spectral coefficient space provides a viable alternative to conventional physical-space approaches.

math.NA↗

arXiv2026

Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves

Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly matched to PDE solutions with strongly heterogeneous complexity across space and space--time, leaving capacity insufficient where the physics is difficult and redundant where it is simple. We introduce physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics. We instantiate this principle with Representation Capacity Adaptation for PDEs (ReCAP), a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging. ReCAP uses responsibility-weighted error indicators and the geometry of residual energy to determine where and how to refine. To limit the disturbance introduced by splitting, we introduce quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation. We further establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Across five challenging 3D and 4D PDE benchmarks against 11 physics-informed solvers, ReCAP achieves the lowest relative $L^2$ error on every problem, reducing error by $10.7\%$--$27.5\%$ relative to the strongest competing result. These results suggest that physics-informed solvers need not merely learn their parameters---they can learn how their representational capacity should be organized.

cs.LG↗

arXiv2026

CARE-SAV: A Conditioning-Aware Random-Feature Framework for Energy-Stable Simulation of Gradient Flows

Gradient-flow models are characterized by an intrinsic energy-dissipation structure, and faithfully preserving this structure at the discrete level is important for stable and reliable long-time simulation. To this end, we develop a Conditioning-Aware Representation Enhancement with Scalar Auxiliary Variable (CARE-SAV) framework, which constructs a compact spatial approximation space from flexible candidate features and evolves the gradient-flow dynamics directly within this space. The resulting fully discrete scheme preserves the discrete energy-dissipation law while providing a flexible alternative to conventional prescribed spatial discretizations. Rigorous analysis establishes the approximation capability, solvability, stability and convergence of the proposed method. Numerical experiments on representative gradient-flow problems demonstrate its accuracy, robustness and computational efficiency. We believe that CARE-SAV could provide a simple, flexible, and computationally efficient paradigm for structure-preserving discretization of gradient-flow problems.

math.NA↗

arXiv2026

C-PINN: A neural network framework based on the Cordès condition for solving linear and fully nonlinear equations in non-divergence form and its applications

In this paper, we propose a novel Physics-Informed Neural Network (PINN) framework based on the Cordès condition for solving both linear and fully nonlinear partial differential equations (PDEs) in non-divergence form, together with their applications. By incorporating the operator structure into the loss function, the proposed method improves the conditioning of the associated optimization problem, thereby enhancing training stability and solution accuracy. The framework is further extended to include Hamilton-Jacobi-Bellman and Monge-Ampère equations, with applications to optimal transport. Numerical experiments demonstrate the effectiveness and robustness of the method, as well as its capability to address high-dimensional problems, highlighting the promise of learning-based approaches for tackling challenging PDEs. Owing to its generality and simplicity, the proposed method is expected to be of broad interest to the scientific and engineering communities.

math.NA↗

arXiv2025

Solving the fully nonlinear Monge-Ampère equation using the Legendre-Kolmogorov-Arnold Network method

In this paper, we propose a novel neural network framework, the Legendre-Kolmogorov-Arnold Network (Legendre-KAN) method, designed to solve fully nonlinear Monge-Ampère equations with Dirichlet boundary conditions. The architecture leverages the orthogonality of Legendre polynomials as basis functions, significantly enhancing both convergence speed and solution accuracy compared to traditional methods. Furthermore, the Kolmogorov-Arnold representation theorem provides a strong theoretical foundation for the interpretability and optimization of the network. We demonstrate the effectiveness of the proposed method through numerical examples, involving both smooth and singular solutions in various dimensions. This work not only addresses the challenges of solving high-dimensional and singular Monge-Ampère equations but also highlights the potential of neural network-based approaches for complex partial differential equations. Additionally, the method is applied to the optimal transport problem in image mapping, showcasing its practical utility in geometric image transformation. This approach is expected to pave the way for further enhancement of KAN-based applications and numerical solutions of PDEs across a wide range of scientific and engineering fields.

math.NA↗