Physics-informed neural networks (PINNs) were recently proposed in [18] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution, while a PDE-induced NN is coupled to the solution NN, and all differential operators are treated using automatic differentiation. Here, we first employ the standard PINN and a stochastic version, sPINN, to solve forward and inverse problems governed by a non-linear advection–diffusion–reaction (ADR) equation, assuming we have some sparse measurements of the concentration field at random or pre-selected locations. Subsequently, we attempt to optimise the hyper-parameters of sPINN by using the Bayesian optimisation method (meta-learning) and compare the results with the empirically selected hyper-parameters of sPINN. In particular, for the first part in solving the inverse deterministic ADR, we assume that we only have a few high-fidelity measurements, whereas the rest of the data is of lower fidelity. Hence, the PINN is trained using a composite multi-fidelity network, first introduced in [12], that learns the correlations between the multi-fidelity data and predicts the unknown values of diffusivity, transport velocity and two reaction constants as well as the concentration field. For the stochastic ADR, we employ a Karhunen–Loève (KL) expansion to represent the stochastic diffusivity, and arbitrary polynomial chaos (aPC) to represent the stochastic solution. Correspondingly, we design multiple NNs to represent the mean of the solution and learn each aPC mode separately, whereas we employ a separate NN to represent the mean of diffusivity and another NN to learn all modes of the KL expansion. For the inverse problem, in addition to stochastic diffusivity and concentration fields, we also aim to obtain the (unknown) deterministic values of transport velocity and reaction constants. The available data correspond to 7spatial points for the diffusivity and 20 space–time points for the solution, both sampled 2000 times. We obtain good accuracy for the deterministic parameters of the order of 1–2% and excellent accuracy for the mean and variance of the stochastic fields, better than three digits of accuracy. In the second part, we consider the previous stochastic inverse problem, and we use Bayesian optimisation to find five hyper-parameters of sPINN, namely the width, depth and learning rate of two NNs for learning the modes. We obtain much deeper and wider optimal NNs compared to the manual tuning, leading to even better accuracy, i.e., errors less than 1% for the deterministic values, and about an order of magnitude less for the stochastic fields.

中文翻译：

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从稀疏测量中学习和元学习随机平流-扩散-反应系统

最近在 [18] 中提出了物理信息神经网络 (PINN) 作为求解偏微分方程 (PDE) 的替代方法。神经网络 (NN) 表示解决方案，而 PDE 诱导的 NN 与解决方案 NN 耦合，并且所有微分算子都使用自动微分处理。在这里，我们首先使用标准 PINN 和随机版本 sPINN 来解决由非线性对流-扩散-反应 (ADR) 方程控制的正向和逆向问题，假设我们对浓度场进行了一些随机或稀疏测量预先选定的位置。随后，我们尝试使用贝叶斯优化方法（元学习）来优化 sPINN 的超参数，并将结果与​​经验选择的 sPINN 超参数进行比较。特别是，对于求解逆确定性 ADR 的第一部分，我们假设我们只有几个高保真测量，而其余数据的保真度较低。因此，PINN 使用复合训练多保真网络，首先在[12]中引入，它学习多保真数据之间的相关性，并预测扩散率、传输速度和两个反应常数以及浓度场的未知值。对于随机 ADR，我们使用 Karhunen-Loève (KL) 展开来表示随机扩散率，并使用任意多项式混沌 (aPC) 来表示随机解。相应地，我们设计了多个 NN 来表示意思是解并分别学习每个 aPC 模式，而我们使用单独的 NN 来表示意思是扩散系数和另一个 NN 来学习 KL 展开的所有模式。对于逆问题，除了随机扩散率和浓度场外，我们还旨在获得传输速度和反应常数的（未知）确定值。可用数据对应 7空间的扩散系数和 20时空解决方案的点，均采样 2000 次。我们获得了 1-2% 数量级的确定性参数的良好精度，以及随机场的均值和方差的出色精度，优于三位数的精度。在第二部分，我们考虑前面的随机逆问题，我们使用贝叶斯优化找到 sPINN 的五个超参数，即两个 NN 的宽度、深度和学习率，用于学习模式。与手动调整相比，我们获得了更深、更广的最优神经网络，从而获得了更好的准确性，即确定性值的误差小于 1%，而随机场的误差大约小一个数量级。