(119f) A Symbolic-Sparse Regression Technique for Data-Driven PDE Learning and Solution Approximation in Chemical Engineering

Dynamic fluid flow, heat transfer, and solution thermodynamics are among the many phenomena in chemical engineering systems that are remarkably well described by partial differential equations (PDEs). Learning accurate PDE models that capture physical and chemical changes from data can be a time-consuming and challenging task. Recent advances in machine learning employ symbolic^1,2 or sparse regression³ to learn PDEs from scarce and noisy data while simultaneously learning solutions to those PDEs, either as numerical solutions or neural-network surrogates. However, compared to analytical solutions, numerical or black-box surrogate solutions offer much less transparency, making it difficult to establish trust and identify or correct inaccurate solutions.

This work demonstrates a novel hybrid approach that merges symbolic regression and sparse regression for the learning of PDEs from data. Additionally, as the PDE is learned, the proposed method yields surrogate analytical solutions, offering higher transparency than other PDE learning methods. To learn PDE models from data, a symbolic regression algorithm, adapted from the highly efficient PySR⁴, searches a symbol space for an expression that satisfies a second order linear PDE. Derivatives of the symbolic expressions proposed by the symbolic regressor are computed using automatic differentiation. As the search occurs, the parameter values of the PDE are updated to best fit the available data, and insignificant terms are removed from the linear PDE, leaving a concise model that represents the observed system.

The approach is demonstrated through three steps. First, using an elliptic PDE with known boundary data, we show that the symbolic regression-based method can learn PDE parameter values from data more accurately than Physics-Informed Neural Networks (PINNs)⁵, while simultaneously yielding analytical solutions. Secondly, starting with a PDE with a known analytical solution, the heat equation, we gradually modify the equation to demonstrate how the method can learn accurate, transparent surrogate solutions for PDEs lacking exact solutions. Finally, we apply the method to learn a simple flow reactor system model from data by eliminating unnecessary terms from a general second-order linear PDE. These examples demonstrate a novel approach to learning PDEs and corresponding analytical solution approximations.

References

1. Cohen, B., Beykal, B. & Bollas, G. Dynamic System Identification from Scarce and Noisy Data Using Symbolic Regression. in 2023 62nd IEEE Conference on Decision and Control (CDC) 3670–3675 (IEEE, 2023). doi:10.1109/CDC49753.2023.10383906.
2. Cohen, B., Beykal, B. & Bollas, G. Physics-Informed Genetic Programming for Discovery of Partial Differential Equations from Scarce and Noisy Data. https://ssrn.com/abstract=4604759.
3. Chen, Z., Liu, Y. & Sun, H. Physics-informed learning of governing equations from scarce data. Nat Commun 12, 6136 (2021).
4. Cranmer, M. Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl. (2023).
5. Raissi, M., Perdikaris, P. & Karniadakis, G. E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378, 686–707 (2019).

Added this sentence to summarize abstract. [CB1]