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

arXiv: https://arxiv.org/abs/2501.06335 · HTML: https://mlsanity.com/paper/2501.06335/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2025.

## Built on by (most cited first, 5 of 5)

- [A Simultaneous Approach for Training Neural Differential-Algebraic Systems of Equations](https://mlsanity.com/paper/2504.04665/) (2025)
- [MathOptAI.jl: Embed trained machine learning predictors into JuMP models](https://mlsanity.com/paper/2507.03159/) (2025)
- [Nonlinear Optimization with GPU-Accelerated Neural Network Constraints](https://mlsanity.com/paper/2509.22462/) (2025)
- [Can AI Follow In Einstein's Footsteps?](https://mlsanity.com/paper/2607.27794/) (2026)
- [Generating adversarial inputs for a graph neural network model of AC power flow](https://mlsanity.com/paper/2602.17975/) (2026)

## Abstract (arXiv)

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.