A Comparison of Strategies to Embed Physics-Informed Neural Networks in Nonlinear Model Predictive Control Formulations Solved via Direct Transcription

TL;DR

This paper benchmarks strategies for integrating physics-informed neural network surrogates into nonlinear model predictive control formulations solved via direct transcription, revealing that external evaluation often outperforms explicit embedding strategies in certain cases.

Contribution

It compares different embedding strategies for physics-informed neural networks in NMPC and provides insights into their computational performance and practical implementation challenges.

Findings

External NN evaluation often outperforms explicit embedding strategies.

Embedding NN as algebraic constraints may not always reduce computation time.

Smooth activation functions can limit the advantages of NN surrogates in NMPC.

Abstract

This study aims to benchmark candidate strategies for embedding neural network (NN) surrogates in nonlinear model predictive control (NMPC) formulations that are subject to systems described with partial differential equations and that are solved via direct transcription (i.e., simultaneous methods). This study focuses on the use of physics-informed NNs and physics-informed convolutional NNs as the internal (surrogate) models within the NMPC formulation. One strategy embeds NN models as explicit algebraic constraints, leveraging the automatic differentiation (AD) of an algebraic modelling language (AML) to evaluate the derivatives. Alternatively, the solver can be provided with derivatives computed external to the AML via the AD routines of the machine learning environment the NN is trained in. The three numerical experiments considered in this work reveal that replacing mechanistic models with NN surrogates may not always offer computational advantages when smooth activation functions are used in conjunction with a local nonlinear solver (e.g., Ipopt), even with highly nonlinear systems. Moreover, in this context, the external function evaluation of the NN surrogates often outperforms the embedding strategies that rely on explicit algebraic constraints, likely due to the difficulty in initializing the auxiliary variables and constraints introduced by explicit algebraic reformulations.