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Free, publicly-accessible full text available June 30, 2026
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In this paper, we propose a new deep learning method for the nonlinear Poisson-Boltzmann problems with applications in computational biology. To tackle the discontinuity of the solution, e.g., across protein surfaces, we approximate the solution by a piecewise mesh-free neural network that can capture the dramatic change in the solution across the interface. The partial differential equation problem is first reformulated as a least-squares physics-informed neural network (PINN)-type problem and then discretized to an objective function using mean squared error via sampling. The solution is obtained by minimizing the designed objective function via standard training algorithms such as the stochastic gradient descent method. Finally, the effectiveness and efficiency of the neural network are validated using complex protein interfaces on various manufactured functions with different frequencies.more » « lessFree, publicly-accessible full text available January 1, 2026
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We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on aggregation-based algebraic multigrid methods with custom smoothers that preserve the coupling information on each coarse level. We prove that, with the proper choice of subspace splitting, we obtain uniform convergence in discretization and physical parameters in the two-level setting. Additionally, we show parameter robustness and scalability with regard to the number of the degrees of freedom of the system on several numerical examples related to the biophysical processes in the brain, namely, the electric signaling in excitable tissue modeled by bidomain, the extracellular-membrane-intracellular (EMI) model, and reduced EMI equations.more » « less
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This paper considered divergence-free basis methods to solve the viscous Stokes equations. A discrete divergence-free subspace was constructed to reduce the saddle point problem of the Stokes problem to a smaller-sized symmetric and positive definite system solely depending on the velocity components. Then, the system could decouple the unknowns in velocity and pressure and solve them independently. However, such a scheme may not ensure an accurate numerical solution to the velocity. In order to obtain satisfactory accuracy, we used a velocity reconstruction technique to enhance the divergence-free scheme to achieve the desired pressure and viscosity robustness. Numerical results were presented to demonstrate the robustness and accuracy of this discrete divergence-free method.more » « less
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