[论文解读] Reduced-PINN: An Integration-Based Physics-Informed Neural Networks for Stiff ODEs

Pouyan Nasiri, Roozbeh Dargazany|arXiv (Cornell University)|Aug 23, 2022

Model Reduction and Neural Networks被引用 6

一句话总结

该论文提出Reduced-PINN，一种物理信息神经网络框架，通过将刚性常微分方程（ODE）重新表述为弱形式积分方程，以克服标准PINNs在刚性系统中训练不稳定和性能差的问题。通过用单个基于积分的损失函数替代多个损失项，Reduced-PINN实现了对刚性化学动力学（包括具有挑战性的ROBER问题）的稳定、精确且计算高效的模拟，其结果与高阶数值求解器（如BDF）相当。

ABSTRACT

Physics-informed neural networks (PINNs) have recently received much attention due to their capabilities in solving both forward and inverse problems. For training a deep neural network associated with a PINN, one typically constructs a total loss function using a weighted sum of different loss terms and then tries to minimize that. This approach often becomes problematic for solving stiff equations since it cannot consider adaptive increments. Many studies reported the poor performance of the PINN and its challenges in simulating stiff chemical active issues with administering conditions of stiff ordinary differential conditions (ODEs). Studies show that stiffness is the primary cause of the failure of the PINN in simulating stiff kinetic systems. Here, we address this issue by proposing a reduced weak-form of the loss function, which led to a new PINN architecture, further named as Reduced-PINN, that utilizes a reduced-order integration method to enable the PINN to solve stiff chemical kinetics. The proposed Reduced-PINN can be applied to various reaction-diffusion systems involving stiff dynamics. To this end, we transform initial value problems (IVPs) to their equivalent integral forms and solve the resulting integral equations using physics-informed neural networks. In our derived integral-based optimization process, there is only one term without explicitly incorporating loss terms associated with ordinary differential equation (ODE) and initial conditions (ICs). To illustrate the capabilities of Reduced-PINN, we used it to simulate multiple stiff/mild second-order ODEs. We show that Reduced-PINN captures the solution accurately for a stiff scalar ODE. We also validated the Reduced-PINN against a stiff system of linear ODEs.

研究动机与目标

• 解决标准PINNs在模拟刚性常微分方程（ODE）时的失败问题，特别是在具有广泛时间尺度差异的化学动力学中。
• 克服标准PINNs中加权和损失函数的局限性，后者因快速与慢速动力学产生的冲突梯度而难以处理刚性问题。
• 通过将初值问题转化为弱形式积分方程，提出统一的单损失公式，降低复杂度并提升训练稳定性。
• 通过在分段时间区间上顺序应用Reduced-PINN，实现对刚性系统的长期模拟。
• 在基准刚性问题上验证该框架，包括非线性的ROBER系统，证明其在准确性和效率上均优于标准PINNs。

提出的方法

• 通过弱形式公式将初值问题（IVPs）转化为等价的积分方程，从而消除显式的ODE和初始条件损失项。
• 基于积分方程的残差构建单一统一的损失函数，替代标准PINNs中的多个损失组件。
• 采用降阶积分方法高效近似积分项，降低计算成本。
• 通过在分段时间区间上顺序应用Reduced-PINN，实现具有更高训练速度和稳定性的长期模拟。
• 通过自动微分强制执行损失函数，训练深度神经网络以近似解轨迹。
• 将后向微分公式（BDF）用作高精度模拟中的对比基准。

实验结果

研究问题

• RQ1 单个基于积分的损失函数能否替代PINNs中的多个损失项，从而提升刚性ODE的训练稳定性？
• RQ2 将ODE转化为弱形式积分方程，是否能实现比标准PINNs更准确、更高效的刚性化学动力学模拟？
• RQ3 Reduced-PINN能否通过将时间域划分为更小区间，成功模拟刚性系统的长期动力学？
• RQ4 Reduced-PINN在具有9个数量级时间尺度差异的基准刚性问题（如ROBER系统）上的表现如何？
• RQ5 Reduced-PINN能否在降低计算成本的同时，实现与高阶数值求解器（如BDF）相当的精度？

主要发现

• Reduced-PINN准确捕捉了刚性标量ODE的解，性能显著优于标准PINNs。
• 对于刚性线性ODE系统，Reduced-PINN与精确解高度一致，验证了其鲁棒性。
• 在ROBER问题——一个具有9个数量级时间尺度差异的基准非线性刚性系统——中，Reduced-PINN以高保真度匹配BDF求解器的结果。
• 与标准PINNs在该类问题上无法收敛相比，该方法实现了计算成本显著降低的精确模拟。
• 通过在分段时间区间上顺序应用Reduced-PINNs，实现了稳定且可扩展的长期模拟。
• 单积分损失公式消除了标准PINNs中常见的梯度不平衡问题，从而实现更可靠的训练。