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Double-activation neural network for solving parabolic equations with time delay

Qiumei Huang, Qiao Zhu

TL;DR

This work tackles solving semilinear parabolic DPDEs with time delay using a novel double-activation neural network (DANN) that augments each neuron with a quadratic term to enhance nonlinear expressivity. It also introduces a piecewise fitting strategy to address derivative discontinuities across the domain, improving accuracy for problems with non-smooth solutions. Theoretical support is provided via a convergence theorem showing the PINN loss can be driven to zero under standard assumptions, and extensive numerical experiments demonstrate that DANN outperforms traditional PINN variants in accuracy and convergence speed, particularly when combined with piecewise fitting. The approach offers a mesh-free, efficient framework for DPDEs with delays, with potential impact on simulations in biology, control, and climate modeling.

Abstract

This paper presents the double-activation neural network (DANN), a novel network architecture designed for solving parabolic equations with time delay. In DANN, each neuron is equipped with two activation functions to augment the network's nonlinear expressive capacity. Additionally, a new parameter is introduced for the construction of the quadratic terms in one of two activation functions, which further enhances the network's ability to capture complex nonlinear relationships. To address the issue of low fitting accuracy caused by the discontinuity of solution's derivative, a piecewise fitting approach is proposed by dividing the global solving domain into several subdomains. The convergence of the loss function is proven. Numerical results are presented to demonstrate the superior accuracy and faster convergence of DANN compared to the traditional physics-informed neural network (PINN).

Double-activation neural network for solving parabolic equations with time delay

TL;DR

This work tackles solving semilinear parabolic DPDEs with time delay using a novel double-activation neural network (DANN) that augments each neuron with a quadratic term to enhance nonlinear expressivity. It also introduces a piecewise fitting strategy to address derivative discontinuities across the domain, improving accuracy for problems with non-smooth solutions. Theoretical support is provided via a convergence theorem showing the PINN loss can be driven to zero under standard assumptions, and extensive numerical experiments demonstrate that DANN outperforms traditional PINN variants in accuracy and convergence speed, particularly when combined with piecewise fitting. The approach offers a mesh-free, efficient framework for DPDEs with delays, with potential impact on simulations in biology, control, and climate modeling.

Abstract

This paper presents the double-activation neural network (DANN), a novel network architecture designed for solving parabolic equations with time delay. In DANN, each neuron is equipped with two activation functions to augment the network's nonlinear expressive capacity. Additionally, a new parameter is introduced for the construction of the quadratic terms in one of two activation functions, which further enhances the network's ability to capture complex nonlinear relationships. To address the issue of low fitting accuracy caused by the discontinuity of solution's derivative, a piecewise fitting approach is proposed by dividing the global solving domain into several subdomains. The convergence of the loss function is proven. Numerical results are presented to demonstrate the superior accuracy and faster convergence of DANN compared to the traditional physics-informed neural network (PINN).
Paper Structure (11 sections, 3 theorems, 32 equations, 11 figures, 5 tables)

This paper contains 11 sections, 3 theorems, 32 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Deep learning for semilinear parabolic DPDEs
  3. PINN for semilinear parabolic DPDEs
  4. Double-activation neual network (DANN)
  5. piecewise fitting
  6. Convergence theorem of the neural network solution
  7. Main results
  8. Preliminaries
  9. Proof of Theorem 3.1
  10. Numerical results
  11. Conclusion

Key Result

Theorem 3.1

Assume (i) the solution of equation (equation) is unique and bounded on domain $\Omega_T$, (ii) the initial function $g(t,x)$ of equation (equation) is bounded on domain ${\Omega_0}$, (iii) the nonlinear terms $h(t,x,u,v)$ is locally Lipschitz continuous in $(u, v)$ with Lipschitz constant that can for some constants $0 \le q_1, q_2, q_3, q_4 < \infty$. Then, for any $\epsilon \ge 0$, there exis

Figures (11)

  • Figure 1: The neuron structure in DNN.
  • Figure 2: The neuron structure in DANN.
  • Figure 3: Training loss curves for solving Example 1.
  • Figure 4: (A) and (B): The figures show the relative $L^2$ errors of $u$ for 28 different architectures across four depth values (2, 4, 6, 8) and seven widths (10, 15, 20, 15, 30, 35, 40) for both PINN and DANN. (C) and (D): The figures show the relative $L^2$ errors of $u$ achieved by varying numbers of $N_I$ (200, 600, 1000, 1400, 1800, 2200) and $N_f$ (2000, 4000, 6000, 8000, 10000, 12000, 15000) for PINN and DANN. Lines indicate mean values and shades represent variances.
  • Figure 5: The exact solution (blue) and the approximate solutions (red) obtained by different models at $t=0.5, 1, 1.5$.
  • ...and 6 more figures

Theorems & Definitions (11)

  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Remark 3.2
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • ...and 1 more