Picard-KKT-hPINN: Enforcing Nonlinear Enthalpy Balances for Physically Consistent Neural Networks

TL;DR

This work tackles the challenge of physically inconsistent neural network surrogates by enforcing nonlinear algebraic constraints, notably enthalpy balances, in surrogate models. It introduces Picard-KKT-hPINN, which extends the KKT-hPINN framework with local projections and variable freezing to achieve exact satisfaction of nonlinear, multiplicatively separable constraints. The method is demonstrated on a methanol synthesis packed-bed reactor, enforcing atomic balances and nonlinear enthalpy balance with machine-level precision, while preserving training efficiency and improving performance in data-scarce regimes. The proposed approach has practical impact for deploying physically consistent neural surrogates in chemical engineering and large-scale process optimization, balancing accuracy, physics compliance, and computational cost.

Abstract

Neural networks are widely used as surrogate models but they do not guarantee physically consistent predictions thereby preventing adoption in various applications. We propose a method that can enforce NNs to satisfy physical laws that are nonlinear in nature such as enthalpy balances. Our approach, inspired by Picard successive approximations method, aims to enforce multiplicatively separable constraints by sequentially freezing and projecting a set of the participating variables. We demonstrate our PicardKKThPINN for surrogate modeling of a catalytic packed bed reactor for methanol synthesis. Our results show that the method efficiently enforces nonlinear enthalpy and linear atomic balances at machine-level precision. Additionally, we show that enforcing conservation laws can improve accuracy in data-scarce conditions compared to vanilla multilayer perceptron.

Figures (2)

• Figure 1: Picard-KKT-hPINN consists of a set of non-trainable layers that can be appended on every NN backbone (e.g., multilayer perceptrons, convolutional neural networks, transformers). Multiplicatively separable constraints (Eq. \ref{['eq:constraints_class']}) are enforced exactly by partitioning the prediction vector (variable freezing). Then, the prediction of the NN is projected onto the feasible space defined by the constraints c. The layers composing Picard-KKT-hPINN are differentiable, hence, the NN is trained end-to-end and the architecture is equivalent at training and inference time.
• Figure 2: Comparison of relative conservation error (RCE) of mass and enthalpy balances (a) and regression performance in data scarcity conditions (b) over the three different models.