Article outline

Purpose. An analytical review of the application of various architectures of Physics-Informed Neural Networks (PINN) for modeling two-phase flows with bubbles.

Methodology. Physics – informed neural networks use constraints in training due to compliance with the laws of physics, the fulfillment of the initial and boundary conditions of the tasks under consideration, and the correspondence of the solution to real experimental data. While solving a problem, a neural network with given hyperparameters minimizes a loss function, which characterizes how much the approximation of the unknown functions vector deviates from the actual solution. Hyperparameters are specified before training and are external parameters of the neural network, determining its configuration: the number of layers, neurons, learning rate, etc. The coefficients before the terms in the global loss function are determined proportionally to the reliability of the corresponding data, ensuring a balanced contribution of each term (differential equations, initial and boundary conditions, and actual data) to the overall optimization problem. Minimization of the loss function is carried out using gradient optimization methods such as ADAM or L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno), which allows iteratively finding the optimal values ​​of weights and biases in the layers of the neural network.

Findings. Various modifications of physically-informed neural networks, their advantages and limitations, are studied for solving problems of bubble dynamics in a two-phase medium.

In particular, distributed PINN architectures are considered, in which the global spatiotemporal domain is divided into cells, each of which has a smaller localized PINN with a single- or two-layer architecture. This avoids the problems of deep neural networks, potentially increasing computational speed. An example of generating a self-adaptive loss function using a Gaussian probability model is given. A comparative analysis of distributed PINN modifications (distributed PINN – DPINN, transfer PINN – TPINN) is presented for the classical problem of a gas bubble rising in a liquid under the action of the Archimedes buoyancy force. DPINNs are shown to have higher accuracy, leading to the conclusion that the use of individual local hypotheses in temporal subdomains is more useful for this problem than sharing learned features across subdomains. The use of local neural networks allows for more accurate prediction of the dynamics of physical processes, similar to mesh refinement in computational fluid dynamics.

This paper discusses the chain architecture of physically-informed neural networks, which is used in cases where it is possible to decompose the original system of equations into individual subsystems, each of which is created with a specialized PINN module, and all the resulting solutions for the subsystems are then sequentially connected into a single chain.

This paper presents the integration of spectral methods into the architecture of physically-informed neural networks, with the solutions decomposition into high-order orthogonal polynomials. The use of these methods allows to obtain a wider range of information on frequency parameters and ensures high prediction accuracy, especially at interfaces and in zones with large velocity gradients.

Value. Unlike traditional computational fluid dynamics (CFD) methods, which require full spatiotemporal resolution at all calculation steps, PINN predicts flow behavior based on limited input data, using fundamental conservation laws built into its loss function. This principal feature significantly reduces dependence on resource-intensive CFD calculations and expensive experimental data. The physical consistency of predictions and conservation laws ensures reliable results even for time steps not represented in the training set, unlike computational fluid dynamics methods, which require labor-intensive tuning of numerical schemes or stabilization methods to achieve similar accuracy. Furthermore, the use of physically-informed neural networks eliminates the need for recalculations when changing resolution, significantly reducing computational costs. Thus, PINN opens up new possibilities for the multiscale analysis of complex flows, significantly reducing the time and cost compared to classical CFD methods.