发表机构
Department of Chemical Engineering, Imperial College London; Department of Operations, University of Lausanne（帝国理工学院化学工程系; 洛桑大学运营系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对动态密度泛函理论中的非局部梯度流方程，开发物理信息神经网络框架。引入修正洛伦兹激活函数和预计算离散算子，经多维度测试，新激活函数加速收敛，框架与参考解吻合且捕捉到梯度流行为，展现求解此类方程的潜力。
AI 中文摘要
我们为动态密度泛函理论（DDFT）中出现的非局部偏微分方程开发了一个物理信息神经网络（PINN）框架。这类方程对标准PINN方法具有挑战性，因其包含非线性、非局部相互作用项及潜在的梯度流结构，常导致收敛慢和优化困难。我们将PINN方法应用于DDFT梯度流方程，引入两个计算组件：一个修正的洛伦兹激活函数，小输入时近似线性表现，输入量增大时趋于零；一个预计算离散算子，用于训练时有效评估非局部卷积项。该方法在一维和二维的四个例子上进行测试。第一个例子有精确稳态解，其余例子将神经网络近似与用连续和间断伽辽金有限元离散化计算的参考解进行验证。通过\(L^1\)、\(L^2\)和\(L^\infty\)误差以及质量守恒和自由能耗散评估准确性和物理一致性。结果表明，相对于标准双曲正切函数，所提出的激活函数加速了收敛，整体框架与参考解保持良好一致并捕捉到预期的梯度流行为。这些发现证明了所提出的PINN框架在求解DDFT中出现的非局部梯度流方程方面的潜力。
英文摘要
We develop a physics-informed neural network (PINN) framework for nonlocal partial differential equations arising in dynamic density functional theory (DDFT). Such equations are challenging for standard PINN methods because they involve nonlinearities, nonlocal interaction terms, and an underlying gradient-flow structure, often leading to slow convergence and difficult optimization. We adapt the PINN methodology to DDFT gradient-flow equations and introduce two computational components: a modified Lorentzian activation function that behaves approximately linearly for small inputs and decays toward zero as the input magnitude increases, and a precomputed discrete operator for evaluating the nonlocal convolution term efficiently during training. The method is tested on four examples in one and two space dimensions. In the first example, the exact stationary solution is known, while in the remaining cases the neural-network approximations are validated against reference solutions computed using continuous and discontinuous Galerkin finite element discretizations. Accuracy and physical consistency are assessed through $L^1$, $L^2$, and $L^\infty$ errors, together with mass conservation and free-energy dissipation. The results show that the proposed activation function accelerates convergence relative to the standard $\tanh$ function, while the overall framework maintains good agreement with the reference solutions and captures the expected gradient-flow behaviour. These findings demonstrate the potential of the proposed PINN framework for solving nonlocal gradient-flow equations arising in DDFT.
