Physics-consistent machine learning: output projection onto physical manifolds

TL;DR

Data-driven surrogates often violate physical laws, limiting reliability. This work introduces a projection-based physics-consistent ML approach that post-processes predictions by projecting them onto manifolds defined by physical constraints, formalized as $\min_{p} \|p - f(x;\Theta)\|^2_W$ subject to $g(x,p)=0$. The method is demonstrated on a spring-mass system and a low-temperature reactive plasma, showing substantial reductions in physics-law violation and improvements in key quantities, while remaining compatible with, and potentially complementary to, PINNs. The results suggest a versatile, scalable tool for fast, reliable surrogates under resource constraints, with robust performance across model complexity and small datasets, and potential for synergistic integration with physics-informed training.

Abstract

Data-driven machine learning models often require extensive datasets, which can be costly or inaccessible, and their predictions may fail to comply with established physical laws. Current approaches for incorporating physical priors mitigate these issues by penalizing deviations from known physical laws, as in physics-informed neural networks, or by designing architectures that automatically satisfy specific invariants. However, penalization approaches do not guarantee compliance with physical constraints for unseen inputs, and invariant-based methods lack flexibility and generality. We propose a novel physics-consistent machine learning method that directly enforces compliance with physical principles by projecting model outputs onto the manifold defined by these laws. This procedure ensures that predictions inherently adhere to the chosen physical constraints, improving reliability and interpretability. Our method is demonstrated on two systems: a spring-mass system and a low-temperature reactive plasma. Compared to purely data-driven models, our approach significantly reduces errors in physical law compliance, enhances predictive accuracy of physical quantities, and outperforms alternatives when working with simpler models or limited datasets. The proposed projection-based technique is versatile and can function independently or in conjunction with existing physics-informed neural networks, offering a powerful, general, and scalable solution for developing fast and reliable surrogate models of complex physical systems, particularly in resource-constrained scenarios.

Figures (6)

• Figure 1: Schematic overview of the approach in this study.a.Artificial NN data-driven model$y = f(x; \Theta)$, where $x$ represents the input vector, $y$ is the predicted output vector, and $\Theta$ denotes the model parameter vector. b.Loss-based PINN model with a regularization term in the loss function, where $\hat{y}_i$ represents the target output vector, $y_i$ represents the predicted output vector, $\mathcal{L}_{Data}$ represents the data loss term between the PINN's predicted outputs and the target data, $\mathcal{R}_j$ represents the residual associated with the $j^{th}$ physical law, $\mathcal{L}_{Physics, j}$ represents the physics loss term regarding the $j^{th}$ physical law imposed to the system, $\mathcal{L}_{Train}$ represents the total training loss function, which is a weighted sum balancing data fitting and adherence to physical constraints, and $\lambda_j$ represents the weight factor given to the $j^{th}$ physical law, with $\lambda = \sum_{j=1}^{M} \lambda_j \leq 1$. c. Formulation and visualization of the projection operation method as a constraint optimization problem, with the projection of the output $y$ of the model onto the manifold defined by the constraint vector $g(x, y) = 0$, where $W$ is a symmetric positive weight matrix.
• Figure 2: Comparative analysis of the models' performance for a given initial condition.a. Predicted and target positions and velocities over time using the four models considered: NN (first column), loss-based PINN (second column), projection method applied to the NN outputs (third column), and projection method applied to the PINN outputs (fourth column). The yellow-shaded areas highlight regions where the NN and loss-based PINN deviate from the target, demonstrating the projection method's corrective performance. The right column provides zoomed-in views of the yellow-shaded regions. b. Predicted and target total energy over time using the four models. c. Bar plot comparing the RMSE for positions, velocities, and energy for the four models considered. The RMSE values are calculated using normalized positions and velocities, while the energy RMSE is computed in Joules, with the initial energy as the target.
• Figure 3: Comparative analysis of the models' performance for several initial conditions. Distributions of root mean square error (RMSE) of the 4 models for each normalized state variable and for energy conservation. Each violin plot contains 100 trajectories and each error bar corresponds to the standard deviation of their RMSEs.
• Figure 4: Steady-state feature estimation in the LTP system.a. Test set results of the four models (NN, loss-based PINN, projected NN, and projected PINN) when predicting the 17 outputs. b. Test set results of the four models when evaluating the compliance with physical laws. c. Relative density of the main species in the mixture at steady-state. d. Bar plot of the three outputs that improved the most when the projection method was applied to the NN predictions. For each output is presented the RMSE of: the NN, the projection when the three constraints are applied simultaneously, the projection when the constraints are applied individually, and the projection when the constraints are applied in pairs.
• Figure 5: Comparative analysis of the model performance before and after applying the projection operation to the NN outputs.a. RMSE of the NN and the NN projection as a function of the model complexity (number of weights and biases in the NN architecture). b. NN and NN projection pressure related trends of the electron density, $n_e$, for $I=30$ mA, $R=12$ mm, and the following NN architectures: (i) [8,8]; (ii) [26,26]; (iii) [1000,1000]. c. RMSE of the NN and the NN projection as a function of the dataset size. d. NN and NN projection pressure related trends of the electron density, $n_e$, for $I=30$ mA, $R=12$ mm, and the following dataset sizes: (i) 200; (ii) 600; (iii) 2500.