Benzhuo Lu

Y. C. Zhou, Benzhuo Lu, Alemayehu A. Gorfe

A continuum electromechanical model is proposed to describe the membrane curvature induced by electrostatic interactions in a solvated protein-membrane system. The model couples the macroscopic strain energy of membrane and the electrostatic solvation energy of the system, and equilibrium membrane deformation is obtained by minimizing the electro-elastic energy functional with respect to the dielectric interface. The model is illustrated with the systems with increasing geometry complexity and captures the sensitivity of membrane curvature to the permanent and mobile charge distributions.

Ruigang Shen, Shi Shu, Ying Yang et al.

We study a two-grid strategy for decoupling the time-dependent Poisson-Nernst-Planck equations describing the mass concentration of ions and the electrostatic potential. The computational system is decoupled to smaller systems by using coarse space solutions at each time level, which can speed up the solution process compared with the finite element method combined with the Gummel iteration. We derive the optimal error estimates in $L^2$ norm for both semi- and fully discrete finite element approximations. Based on the a priori error estimates, the error estimates in $H^1$ norm are presented for the two-grid algorithm. The theoretical results indicate this decoupling method can retain the same accuracy as the finite element method. Numerical experiments including the Poisson-Nernst-Planck equations for an ion channel show the efficiency and effectiveness of the decoupling two-grid method.

Tiantian Liu, Minxin Chen, Benzhuo Lu

Recent developments for mathematical modeling and numerical simulation of biomolecular systems raise new demands for qualified, stable, and efficient surface meshing, especially in implicit-solvent modeling. In our former work, we have developed an algorithm for manifold triangular meshing for large Gaussian molecular surfaces, TMSmesh. In this paper, we present new algorithms to greatly improve the meshing efficiency and qualities, and implement into a new program version, TMSmesh 2.0. In TMSmesh 2.0, in the first step, a new adaptive partition and estimation algorithm is proposed to locate the cubes in which the surface are approximated by piecewise trilinear surface with controllable precision. Then, the piecewise trilinear surface is divided into single valued pieces by tracing along the fold curves, which ensures that the generated surface meshes are manifolds. Numerical test results show that TMSmesh 2.0 is capable of handling arbitrary sizes of molecules and achieves ten to hundreds of times speedup over the previous algorithm. The result surface meshes are manifolds and can be directly used in boundary element method (BEM) and finite element method (FEM) simulation. The binary version of TMSmesh 2.0 is downloadable at the web page http://lsec.cc.ac.cn/~lubz/Meshing.html.

Heng Wu, Junjie Wang, Benzhuo Lu

Neural operator learning directly constructs the mapping relationship from the equation parameter space to the solution space, enabling efficient direct inference in practical applications without the need for repeated solution of partial differential equations (PDEs) - an advantage that is difficult to achieve with traditional numerical methods. In this work, we find that explicitly decoupling linear and nonlinear effects within such operator mappings leads to markedly improved learning efficiency. This yields a novel network structure, namely the Linear-Nonlinear Fusion Neural Operator (LNF-NO), which models operator mappings via the multiplicative fusion of a linear component and a nonlinear component, thus achieving a lightweight and interpretable representation. This linear-nonlinear decoupling enables efficient capture of complex solution features at the operator level while maintaining stability and generality. LNF-NO naturally supports multiple functional inputs and is applicable to both regular grids and irregular geometries. Across a diverse suite of PDE operator-learning benchmarks, including nonlinear Poisson-Boltzmann equations and multi-physics coupled systems, LNF-NO is typically substantially faster to train than Deep Operator Networks (DeepONet) and Fourier Neural Operators (FNO), while achieving comparable or better accuracy in most cases. On the tested 3D Poisson-Boltzmann case, LNF-NO attains the best accuracy among the compared models and trains approximately 2.7x faster than a 3D FNO baseline.

Haochen Wu, Heng Wu, Benzhuo Lu

For linear partial differential equations with known fundamental solutions, this work introduces a novel operator learning framework that relies exclusively on domain boundary data, including solution values and normal derivatives, rather than full-domain sampling. By integrating the previously developed Mathematical Artificial Data (MAD) method, which enforces physical consistency, all training data are synthesized directly from the fundamental solutions of the target problems, resulting in a fully data-driven pipeline without the need for external measurements or numerical simulations. We refer to this approach as the Mathematical Artificial Data Boundary Neural Operator (MAD-BNO), which learns boundary-to-boundary mappings using MAD-generated Dirichlet-Neumann data pairs. Once trained, the interior solution at arbitrary locations can be efficiently recovered through boundary integral formulations, supporting Dirichlet, Neumann, and mixed boundary conditions as well as general source terms. The proposed method is validated on benchmark operator learning tasks for two-dimensional Laplace, Poisson, and Helmholtz equations, where it achieves accuracy comparable to or better than existing neural operator approaches while significantly reducing training time. The framework is naturally extensible to three-dimensional problems and complex geometries.

Heng Wu, Benzhuo Lu

Machine learning has emerged as a transformative tool for solving differential equations (DEs), yet prevailing methodologies remain constrained by dual limitations: data-driven methods demand costly labeled datasets while model-driven techniques face efficiency-accuracy trade-offs. We present the Mathematical Artificial Data (MAD) framework, a new paradigm that integrates physical laws with data-driven learning to facilitate large-scale operator discovery. By exploiting DEs' intrinsic mathematical structure to generate physics-embedded analytical solutions and associated synthetic data, MAD fundamentally eliminates dependence on experimental or simulated training data. This enables computationally efficient operator learning across multi-parameter systems while maintaining mathematical rigor. Through numerical demonstrations spanning 2D parametric problems where both the boundary values and source term are functions, we showcase MAD's generalizability and superior efficiency/accuracy across various DE scenarios. This physics-embedded-data-driven framework and its capacity to handle complex parameter spaces gives it the potential to become a universal paradigm for physics-informed machine intelligence in scientific computing.

Ying Yang, Benzhuo Lu, Yan Xie

Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. Two kinds of two-grid finite element algorithms are proposed to decouple the steady-state Poisson-Nernst-Planck equations by coarse grid finite element approximations. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid algorithms for solving Poisson-Nernst-Planck equations.