Machine Learning-Augmented Partial Differential Equation Solvers for High-Fidelity Engineering Design Optimization

Abstract

Solution of multiphysics problems in aerospace, energy and manufacturing systems involve engineering design optimization, where partial differential equations (PDEs) prevail. Nevertheless, the dimensionality of parameter spaces and repetitive calculations in optimization loops mean that the traditional solvers: finite element, finite difference, and spectral methods are computationally expensive. This paper presents a machine learning (ML)-enhanced solver of PDEs that can be used to optimize the design of engineering applications at high-fidelity. The framework is a combination of neural operator architecture, surrogate modeling, and physics-informed learning to improve solution efficiency without compromising accuracy. The algorithm combines PDE solvers at baseline to generate data with ML surrogates, which model solution operators and objective functions. Physics-informed loss is used to satisfy governing equations and adaptive sampling plans are used to improve performance of surrogates in important design regions. The approach is seen to be effective in benchmark case studies in structural mechanics, thermal conduction, and fluid dynamics. In all these domains, the ML-enhanced framework can reduce up to 65 percent of the computational time in comparison to traditional approaches, and the error rate always remains less than 2 percent. These findings indicate a promise of ML-augmented PDE solvers to greatly decrease design cycle times without fidelity loss. The proposed framework provides a way of next-generation automation in the design of engineering models; this is because it can provide scalable, multi-objective optimization to computationally intensive computational programs. Future developments will cover quantification of uncertainty, multi-scale modeling and large scale implementation in the distributed high-performance computing infrastructure.