Given the existence of various forward and inverse problems in combustion
studies and applications that necessitate distinct methods for resolution, a
framework to solve them in a unified way is critically needed. A promising
approach is the integration of machine learning methods with governing
equations of combustion systems, which exhibits superior generality and
few-shot learning ability compared to purely data-driven methods. In this work,
the FlamePINN-1D framework is proposed to solve the forward and inverse
problems of 1D laminar flames based on physics-informed neural networks. Three
cases with increasing complexity have been tested: Case 1 are
freely-propagating premixed (FPP) flames with simplified physical models, while
Case 2 and Case 3 are FPP and counterflow premixed (CFP) flames with detailed
models, respectively. For forward problems, FlamePINN-1D aims to solve the
flame fields and infer the unknown eigenvalues (such as laminar flame speeds)
under the constraints of governing equations and boundary conditions. For
inverse problems, FlamePINN-1D aims to reconstruct the continuous fields and
infer the unknown parameters (such as transport and chemical kinetics
parameters) from noisy sparse observations of the flame. Our results strongly
validate these capabilities of FlamePINN-1D across various flames and working
conditions. Compared to traditional methods, FlamePINN-1D is differentiable and
mesh-free, exhibits no discretization errors, and is easier to implement for
inverse problems. The inverse problem results also indicate the possibility of
optimizing chemical mechanisms from measurements of laboratory 1D flames.
Furthermore, some proposed strategies, such as hard constraints and thin-layer
normalization, are proven to be essential for the robust learning of
FlamePINN-1D. The code for this paper is partially available at
this https URL

提出了 FlamePINN-1D 框架，结合燃烧系统的控制方程与机器学习方法解决一维层流火焰的正向和逆向问题，并通过实验证实了该框架对各种火焰及工况的有效性以及优于传统方法的特点。

FlamePINN-1D：1D 层流火焰的物理信息神经网络求解正向和反向问题

FlamePINN-1D: Physics-informed neural networks to solve forward and  inverse problems of 1D laminar flames

The emergence of neural networks constrained by physical governing equations
has sparked a new trend in deep learning research, which is known as
Physics-Informed Neural Networks (PINNs). However, solving high-dimensional
problems with PINNs is still a substantial challenge, the space complexity
brings difficulty to solving large multidirectional problems. In this paper, a
novel PINN framework to quickly predict several three-dimensional Terzaghi
consolidation cases under different conditions is proposed. Meanwhile, the loss
functions for different cases are introduced, and their differences in
three-dimensional consolidation problems are highlighted. The tuning strategies
for the PINNs framework for three-dimensional consolidation problems are
introduced. Then, the performance of PINNs is tested and compared with
traditional numerical methods adopted in forward problems, and the coefficients
of consolidation and the impact of noisy data in inverse problems are
identified. Finally, the results are summarized and presented from
three-dimensional simulations of PINNs, which show an accuracy rate of over 99%
compared with ground truth for both forward and inverse problems. These results
are desirable with good accuracy and can be used for soil settlement
prediction, which demonstrates that the proposed PINNs framework can learn the
three-dimensional consolidation PDE well.
Keywords: Three-dimensional Terzaghi consolidation; Physics-informed neural
networks (PINNs); Forward problems; Inverse problems; soil settlement

提出了一种新颖的基于物理信息的神经网络（PINNs）框架，用于快速预测不同条件下的几个三维 Terzaghi 固结情况，并引入了不同情况下的损失函数，在三维固结问题中突出了它们的差异。介绍了三维固结问题中 PINNs 框架的调优策略，并将其性能与前向问题中采用的传统数值方法进行了测试和比较，确定了固结系数和逆问题中噪声数据的影响。最后，通过针对三维固结问题的模拟总结和展示了 PINNs 的结果，与地面实况相比，准确率超过 99％，在正向和逆向问题上均表现出理想的准确性，可用于土壤沉降预测，证明了所提出的 PINNs 框架能够很好地学习三维固结偏微分方程。

基于物理信息的深度学习解决三维 Terzaghi 固结方程：正向和反向问题

Physics-informed Deep Learning to Solve Three-dimensional Terzaghi  Consolidation Equation: Forward and Inverse Problems

We employ adaptive activation functions for regression in deep and
physics-informed neural networks (PINNs) to approximate smooth and
discontinuous functions as well as solutions of linear and nonlinear partial
differential equations. In particular, we solve the nonlinear Klein-Gordon
equation, which has smooth solutions, the nonlinear Burgers equation, which can
admit high gradient solutions, and the Helmholtz equation. We introduce a
scalable hyper-parameter in the activation function, which can be optimized to
achieve best performance of the network as it changes dynamically the topology
of the loss function involved in the optimization process. The adaptive
activation function has better learning capabilities than the traditional one
(fixed activation) as it improves greatly the convergence rate, especially at
early training, as well as the solution accuracy. To better understand the
learning process, we plot the neural network solution in the frequency domain
to examine how the network captures successively different frequency bands
present in the solution. We consider both forward problems, where the
approximate solutions are obtained, as well as inverse problems, where
parameters involved in the governing equation are identified. Our simulation
results show that the proposed method is a very simple and effective approach
to increase the efficiency, robustness and accuracy of the neural network
approximation of nonlinear functions as well as solutions of partial
differential equations, especially for forward problems.

采用自适应激活函数进行深度和物理知识神经网络中的回归以逼近光滑和不连续函数以及线性和非线性偏微分方程的解。该方法通过在激活函数中引入可扩展的超参数，并考虑前向问题和反向问题，显着提高了神经网络学习能力和近似解决方案的收敛速度、准确度和鲁棒性。