Session: 36-09 Machine Learning & Artificial Intelligence Methods - Part 4

Submission Number: 178434

Physics-Informed Convolutional Neural Networks for Solving Quasi-1D Adjoint Equations 

The pursuit of extreme performance in aero engines highlights the limitations of experience-based manual design approaches, underscoring the critical advantages of numerical simulation-based design optimization. Among these methods, the adjoint method is characterized by its gradient computation cost remaining nearly independent of the number of design variables, offering significant computational efficiency and leading to its widespread adoption.

In practical applications, however, the adjoint method must be embedded within an iterative cycle comprising flow field solution, adjoint field solution, and gradient-based optimization. Despite the computational efficiency of a single iteration, this overall process remains time-consuming as it must be repeated numerous times. Consequently, accelerating this iterative process, particularly by reducing the time required for adjoint field computation, is crucial for enhancing optimization efficiency.

Physics-informed neural networks (PINNs) present a promising alternative. By incorporating the residuals of the governing physical equations into the loss function, PINNs reduce reliance on large labeled datasets and improve model generalizability. Although applying PINNs directly to solve high-Reynolds-number flow fields is challenging due to strong nonlinearities, the adjoint equations—generally a linear system—possess a simpler solution space. This linearity may make them more amenable to learning by neural networks, thereby circumventing the difficulties associated with highly nonlinear flow solutions.

This study investigates the use of a physics-informed convolutional neural network (PICNN) to solve the adjoint equations of the quasi-one-dimensional Euler equations. Focusing on flow through one-dimensional nozzles with variable cross-sectional areas, the objective is to develop an end-to-end surrogate model that maps steady flow fields to their corresponding adjoint fields. Specifically, Latin Hypercube Sampling is employed to generate a diverse set of geometric configurations. The steady flow field solutions serve as the network input, while the corresponding adjoint fields, obtained via traditional methods, constitute the target output. A U-Net architecture, featuring four down-sampling and up-sampling layers, is constructed. Physical constraints are implemented by incorporating the discretized residuals of both the continuous adjoint equations and the discrete adjoint equations into the loss function.

The study compares three training strategies: a purely data-driven approach, a hybrid approach combining limited data with physics-driven constraints, and a purely physics-driven approach. Results indicate that the hybrid strategy achieves the best balance between prediction accuracy and generalizability, outperforming the other two methods. In an optimization case study, this surrogate model successfully replaced the traditional adjoint solver, enabling a rapid map from flow fields to adjoint fields and significantly reducing the time per optimization iteration.

This work validates the feasibility of using physics-informed convolutional neural networks for efficiently solving linear adjoint equations. It lays the groundwork for future extension to two- and three-dimensional adjoint problems, with the long-term goal of enabling efficient adjoint-based turbomachinery optimization using physics-informed neural networks.

Presenting Author: Bowen Li Tsinghua University

Presenting Author Biography: Li Bowen is a Ph.D. Candidate in Aeronautical and Astronautical Science and Technology at Tsinghua University. His research focuses on turbomachinery design, CFD development, and adjoint methods. This study is a key part of his dissertation, exploring the use of physics-informed neural networks to efficiently solve adjoint equations, with the goal of enabling rapid and high-fidelity design optimization.

Authors:

Bowen Li Tsinghua University
Xin Li Tsinghua University
Lucheng Ji Tsinghua University