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Runge-Kutta Random Feature Method for Solving Multiphase Flow Problems of Cells

Yangtao Deng, Qiaolin He

TL;DR

This work introduces the Runge-Kutta Random Feature Method (RK-RFM), a mesh-free framework that solves strongly nonlinear, time-dependent multiphase cell-flow PDEs by combining a space-based random feature representation with an explicit RK time integrator. The method enables high accuracy in both space and time, leverages parallelization and manual derivative calculation to reduce computational cost, and provides theoretical error estimates that decompose into RFM-approximation and RK-discretization components. Numerical convergence tests and large-scale multiphase cell-flow simulations validate the approach, revealing second-order convergence and realistic cell dynamics, including activity-threshold behavior. Overall, RK-RFM offers an efficient, scalable solver for complex tissue mechanics problems with potential applicability to other nonlinear PDE systems.

Abstract

Cell collective migration plays a crucial role in a variety of physiological processes. In this work, we propose the Runge-Kutta random feature method to solve the nonlinear and strongly coupled multiphase flow problems of cells, in which the random feature method in space and the explicit Runge-Kutta method in time are utilized. Experiments indicate that this algorithm can effectively deal with time-dependent partial differential equations with strong nonlinearity, and achieve high accuracy both in space and time. Moreover, in order to improve computational efficiency and save computational resources, we choose to implement parallelization and non-automatic differentiation strategies in our simulations. We also provide error estimates for the Runge-Kutta random feature method, and a series of numerical experiments are shown to validate our method.

Runge-Kutta Random Feature Method for Solving Multiphase Flow Problems of Cells

TL;DR

This work introduces the Runge-Kutta Random Feature Method (RK-RFM), a mesh-free framework that solves strongly nonlinear, time-dependent multiphase cell-flow PDEs by combining a space-based random feature representation with an explicit RK time integrator. The method enables high accuracy in both space and time, leverages parallelization and manual derivative calculation to reduce computational cost, and provides theoretical error estimates that decompose into RFM-approximation and RK-discretization components. Numerical convergence tests and large-scale multiphase cell-flow simulations validate the approach, revealing second-order convergence and realistic cell dynamics, including activity-threshold behavior. Overall, RK-RFM offers an efficient, scalable solver for complex tissue mechanics problems with potential applicability to other nonlinear PDE systems.

Abstract

Cell collective migration plays a crucial role in a variety of physiological processes. In this work, we propose the Runge-Kutta random feature method to solve the nonlinear and strongly coupled multiphase flow problems of cells, in which the random feature method in space and the explicit Runge-Kutta method in time are utilized. Experiments indicate that this algorithm can effectively deal with time-dependent partial differential equations with strong nonlinearity, and achieve high accuracy both in space and time. Moreover, in order to improve computational efficiency and save computational resources, we choose to implement parallelization and non-automatic differentiation strategies in our simulations. We also provide error estimates for the Runge-Kutta random feature method, and a series of numerical experiments are shown to validate our method.

Paper Structure

This paper contains 11 sections, 3 theorems, 57 equations, 11 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Random Feature Method and Multiphase Flow Problem of Cells
  3. Random Feature Method for Static Problems
  4. Multiphase Flow Problem of Cells
  5. Runge-Kutta Random Feature Method and Error Estimation
  6. Runge-Kutta Random Feature Method
  7. Error Estimates of Runge-Kutta Random Feature Method
  8. Numerical experiments
  9. Convergence Test
  10. Multiphase Flow Problem of Cells
  11. Conclusions and remarks

Key Result

Lemma 3.1

Let $\sigma$ be a bounded nonconstant piecewise continuous activation function, $d_{\phi}=1$, $G_{M}(\boldsymbol{x})$ be of the form eq:approximatesolution, and $M = M_pJ_n$ denotes the degree of freedom. Then, for any $f \in C(\Omega)$, there exists a sequence of $G_{M}(\boldsymbol{x})$, such that

Figures (11)

  • Figure 1: The architecture of of Algorithm \ref{['alg:Algorithm 1']}.
  • Figure 2: The arrangement of subdomains and the distribution of collocation points in subdomain $\Omega_{n}$. (a) The blue lines and red lines represent the boundaries and the interfaces between subdomains, respectively. (b) The green points and yellow points represent the boundary points and interior points in each subdomain, respectively.
  • Figure 3: Relative $L^2$ error for \ref{['eq:exactsolution']} obtained by the Algorithm \ref{['alg:Algorithm 1']} with $\Delta t=$ 5E-1, 5E-2, 5E-3, 5E-4 and 5E-5.
  • Figure 4: Relative $L^2$ error for \ref{['eq:exactsolution']} obtained by the Algorithm \ref{['alg:Algorithm 1']} with $J_n=$ 50, 100, 150 and 200.
  • Figure 5: Relative $L^2$ error for \ref{['eq:exactsolution']} obtained by the Algorithm \ref{['alg:Algorithm 1']} with $M_p=$ 1$\times$1, 2$\times$2, 3$\times$3 and 4$\times$4.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof
  • Remark 3