arXiv subjects

Jiaxiong Hao

Publications and source records attributed to Jiaxiong Hao.

3 recordsLinked to original sources

arXiv2026

Sparsely connected neural network representation of Lagrange finite element function

We construct a mesh-induced sparsely connected neural network framework that exactly reproduces arbitrary-order Lagrange finite element spaces over simplicial meshes. Unlike conventional black-box neural surrogates, the proposed network architecture is fully dictated by finite element discretization rules: local computations stem from simplex geometry and barycentric coordinate transformations, while global consistency is enforced through shared degrees of freedom. For linear Lagrange elements, local basis functions are directly implemented via affine barycentric layers, and high-order polynomial bases are explicitly decomposed into barycentric product compositions realized by specially designed $\mathrm{ReLU}^p$ modules. Equipped with element indicator branches and multiplication units, these modular local components are globally assembled into a sparsely connected neural network whose function space coincides exactly with the target finite element space, thereby inheriting the complete classical finite element approximation theory. By prescribing customized backward differentiation rules for piecewise activations, function values and their spatial gradients can be simultaneously extracted via automatic differentiation within a unified computational graph, eliminating the separate gradient calculation subroutines required in standard finite element implementations. Numerical experiments verify the accuracy of the neural network representation of Lagrange finite elements. Furthermore, by virtue of the intrinsic mesh-free nature of this neural network representation, finite element functions can be interpolated between non-matching meshes, and the proposed scheme can be applied to adaptive finite element methods for solving parabolic partial differential equations. An open-source code implementation of the proposed architecture is made publicly available.

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arXiv2026

Neural network-enhanced $hr$-adaptive finite element algorithm for parabolic equations

In this paper, we propose a novel $hr$-adaptive finite element method, enhanced by neural networks, for parabolic equations. The main challenge of the conventional $h$-adaptive finite element method is interpolating the finite element solution from the previous step in the updated mesh. The interpolation dependent on the new mesh must be recomputed at each adaptive iteration, resulting in high computational costs. The new approach addresses this challenge by introducing a neural network to construct a mesh-free surrogate of the previous step finite element solution. Since the neural network is mesh-free, it only requires training once per time step, with its parameters initialized using the minimizer of the previous time step. This approach effectively overcomes the interpolation challenges associated with non-nested meshes in computation, making node insertion and movement more convenient and efficient. The new algorithm also emphasizes SIZE and GENERATE, allowing each refinement to roughly double the number of mesh nodes of the previous iteration and then redistribute them to form a new mesh that effectively captures the singularities. It significantly reduces the time required for repeated refinement of the conventional methods and achieves the desired accuracy in no more than seven space-adaptive iterations per time step. Numerical experiments confirm the efficiency of the proposed algorithm in capturing dynamic changes of singularities. The code is made publicly available on GitHub.

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arXiv2026

DataTransfer: Neural network based interpolation across non-nested meshes

In mesh-based numerical simulations, the interpolation of mesh-defined functions across different meshes is a critical task, and achieving high-precision interpolation is of great significance for improving the computational efficiency and numerical stability of algorithms. This paper proposes neural network based function mapping model across meshes, wherein the interpolation process is reformulated as a data-driven regression problem over scattered function data. Conventional interpolation and projection-based approaches are highly dependent on mesh connectivity and corresponding geometric properties, which renders such methods computationally costly and sensitive to mismatches between source and target meshes. The proposed method constructs a neural network approximator using nodal function values on the source mesh to obtain a global representation of the function, which can then be interpolated onto any other meshes. To investigate the network architectural impacts on model performance, three representative feedforward network structures are numerically compared in this work: multi-layer perceptrons, extreme learning machines, and network incorporating radial basis function hidden units. The results reveal distinct trade-offs among accuracy, computational efficiency and model robustness, among which the radial basis function-based network achieves the most desirable overall performance balance, enabling fast and precise function calculation. Numerical experiments conducted on non-nested meshes validate the efficacy of the proposed model in both function interpolation and cross-mesh data transmission tasks.

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