Shared Symbolic Backbones for Physically Consistent Multi-Output Symbolic Regression
Introduced MO-SB-NESR method to improve physical consistency in multi-output symbolic regression.
Key Findings
Methodology
The paper presents a novel symbolic regression method, MO-SB-NESR, for handling multi-output systems. It ensures cross-output consistency through a shared symbolic backbone. The model structure evolves through mutation and crossover, while continuous parameters are adjusted via gradient descent.
Key Results
- In the Langmuir-Hinshelwood benchmark, MO-SB-NESR successfully identified shared factors, reducing inconsistency.
- In the Van de Vusse benchmark, independent symbolic regression performed comparably to the coupled model.
- In the hydrothermal liquefaction yield case, MO-SB-NESR demonstrated its capability as a shared structure extractor.
Significance
This research addresses the challenge of maintaining physical consistency in independent symbolic regression, especially when shared factors are weakly identifiable from data. It provides a structured mechanism extractor for engineering applications.
Technical Contribution
Compared to existing methods, this approach offers a new structure-sharing mechanism, capable of identifying and reusing symbolic units across multiple outputs. It innovates in sparsity and sharing mechanisms.
Novelty
This is the first to address physical consistency in multi-output symbolic regression through shared symbolic backbones. It offers higher structural compactness compared to independent regression methods.
Limitations
- When each output is identifiable, the coupled model may not outperform independent regression.
- The method is seed-sensitive, requiring multiple runs for optimal results.
Future Work
Future research can explore applying this method to more complex systems and study its adaptability across different fields.
AI Executive Summary
Symbolic regression is typically used to generate analytical expressions, but it has limitations in handling multi-output systems. Independent symbolic regression can produce models that are difficult to interpret. This paper introduces a novel neuro-evolutionary symbolic regression method, MO-SB-NESR, to address this issue. The method ensures cross-output consistency through shared symbolic backbones and evolves model structures through mutation and crossover. Experimental results show that the method performs well in Langmuir-Hinshelwood and hydrothermal liquefaction yield cases, successfully identifying shared factors and reducing inconsistency. While the coupled model may not always outperform independent regression, the method's value lies in its structural compactness and consistency. Future research can explore applying this method to more complex systems.
Deep Analysis
Background
Symbolic regression is used to generate analytical expressions but is typically applied to single outputs. This presents limitations in process systems where state variables are often coupled through shared physical parameters. Independent symbolic regression can produce models that are difficult to interpret.
Core Problem
Independent symbolic regression struggles to maintain physical consistency in multi-output systems, especially when shared factors are weakly identifiable. A method is needed to ensure cross-output consistency.
Innovation
Introduced MO-SB-NESR method to address consistency issues in multi-output symbolic regression through shared symbolic backbones. The method evolves model structures through mutation and crossover and adjusts continuous parameters via gradient descent.
Methodology
- �� Search for shared symbolic backbone: discover and reuse symbolic units.
- �� Evolve model structure through mutation and crossover.
- �� Adjust continuous parameters via gradient descent.
- �� Evaluate cross-output consistency.
Experiments
Assessed using benchmarks with known ground truth and hydrothermal liquefaction yield case. Compared performance of coupled model with independent symbolic regression.
Results
In Langmuir-Hinshelwood benchmark, MO-SB-NESR successfully identified shared factors. In Van de Vusse benchmark, independent symbolic regression performed comparably to the coupled model.
Applications
The method can be used for process system modeling in chemical engineering, ensuring physical consistency in models.
Limitations & Outlook
When each output is identifiable, the coupled model may not outperform independent regression. The method is seed-sensitive, requiring multiple runs for optimal results.
Plain Language Accessible to non-experts
Imagine you're cooking in a kitchen. Independent symbolic regression is like preparing ingredients separately for each dish, while the MO-SB-NESR method finds a shared base ingredient that can be used across multiple dishes. This not only saves time but ensures consistent flavor in each dish.
ELI14 Explained like you're 14
Imagine you're playing a game with different levels. Independent symbolic regression is like learning new rules for each level, while the MO-SB-NESR method finds a shared strategy that can be used across multiple levels. This makes the game easier and ensures you score well in each level!
Glossary
Symbolic Regression
A machine learning method for generating analytical expressions.
Used for modeling multi-output systems.
Neuro-Evolutionary
A method combining neural networks and evolutionary algorithms.
Used to optimize model structure and parameters.
Shared Backbone
A structure of shared symbolic units.
Ensures cross-output consistency.
Gradient Descent
An optimization algorithm used to adjust model parameters.
Used to adjust continuous parameters.
Mutation
An evolutionary algorithm operation used to change model structure.
Used to evolve model structure.
Open Questions Unanswered questions from this research
- 1 How to apply MO-SB-NESR method to more complex systems remains to be explored.
- 2 The adaptability of the method across different fields is not fully understood.
Applications
Immediate Applications
Chemical Engineering Modeling
MO-SB-NESR can be used for process system modeling in chemical engineering, ensuring physical consistency in models.
Long-term Vision
Cross-Field Applications
Explore applying this method in other fields like bioengineering and environmental science.
Abstract
Symbolic regression provides analytical expressions, but it is usually applied one output at a time. This is limiting in process systems, where state variables are often coupled through shared physical parameters. Independent symbolic regression can give accurate individual equations that are difficult to interpret as one model. We present a neuro-evolutionary symbolic regression method for coupled multi-output systems. The method searches for a shared symbolic backbone: a set of latent symbolic units that is discovered once and reused by several outputs through sparse additive or multiplicative read-outs. The discrete model structure is evolved by mutation and crossover, whereas the continuous parameters are tuned by gradient descent and inherited by the offspring. The method is assessed on a set of benchmarks with known ground truth and on a hydrothermal liquefaction yield case. The results show that coupling is not a general route to lower prediction error. Its main contribution is the enforcement and diagnosis of cross-output consistency when a physically shared factor is embedded in a latent expression and is weakly identifiable from the data. This occurs for Langmuir-Hinshelwood and site-coverage denominators, for which independent PySR does not close the consistency gap or recover the same shared form. Conversely, when each output is already identifiable, as in the Van de Vusse benchmark, independent symbolic regression matches or improves the coupled model. The proposed framework, rather than a general purpose predictor, is a structured shared-mechanism extractor. Its value is highest when the target structure is sparse, shared, weakly identifiable or constrained by closure.