Differential equations offer a foundational yet powerful framework for modeling interactions within complex dynamic systems and are widely applied across numerous scientific fields. One common challenge in this area is estimating the unknown parameters of these dynamic relationships. However, traditional numerical optimization methods rely on the selection of initial parameter values, making them prone to local optima. Meanwhile, deep learning and Bayesian methods require training models on specific differential equations, resulting in poor versatility. This paper reformulates the parameter estimation problem of differential equations as an optimization problem by introducing the concept of particles from the particle swarm optimization algorithm. Building on reinforcement learning-based particle swarm optimization (RLLPSO), this paper proposes a novel method, DERLPSO, for estimating unknown parameters of differential equations. We compared its performance on three typical ordinary differential equations with the state-of-the-art methods, including the RLLPSO algorithm, traditional numerical methods, deep learning approaches, and Bayesian methods. The experimental results demonstrate that our DERLPSO consistently outperforms other methods in terms of performance, achieving an average Mean Square Error of 1.13e-05, which reduces the error by approximately 4 orders of magnitude compared to other methods. Apart from ordinary differential equations, our DERLPSO also show great promise for estimating unknown parameters of partial differential equations. The DERLPSO method proposed in this paper has high accuracy, is independent of initial parameter values, and possesses strong versatility and stability. This work provides new insights into unknown parameter estimation for differential equations.

翻译：微分方程为复杂动态系统内部的相互作用提供了一个基础且强大的建模框架，在众多科学领域得到广泛应用。该领域的一个常见挑战在于估计这些动态关系中的未知参数。然而，传统的数值优化方法依赖于初始参数值的选择，容易陷入局部最优。同时，深度学习和贝叶斯方法需要在特定微分方程上训练模型，导致其泛化能力较差。本文通过引入粒子群优化算法中的粒子概念，将微分方程的参数估计问题重新表述为一个优化问题。在基于强化学习的粒子群优化算法（RLLPSO）基础上，本文提出了一种用于估计微分方程未知参数的新方法DERLPSO。我们在三个典型的常微分方程上将其性能与最先进的方法进行了比较，包括RLLPSO算法、传统数值方法、深度学习方法和贝叶斯方法。实验结果表明，我们的DERLPSO在性能上始终优于其他方法，平均均方误差达到1.13e-05，与其他方法相比误差降低了约4个数量级。除了常微分方程，我们的DERLPSO在估计偏微分方程的未知参数方面也展现出巨大潜力。本文提出的DERLPSO方法具有高精度、不依赖于初始参数值、并具备强大的泛化能力和稳定性。这项工作为微分方程的未知参数估计提供了新的思路。