Automated Scientific Formula Discovery via Symbolic Regression and Theorem Proving

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Solution Overview

Problem

Current methods for discovering scientific formulas from experimental data lack the ability to generate models that are both accurate and derivable from a set of axioms, with existing automated theorem provers facing challenges in computational complexity and derivability, especially when dealing with arithmetic and calculus operators.

Innovation Solution

A system that combines machine learning with a reasoning model to generate candidate formulas from data points, process them through a symbolic regression module, and then uses a reasoning engine to determine their compatibility with a background theory, providing a performance metric and error analysis to identify meaningful and valid new scientific formulas.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Reliability

If automated theorem provers are used to verify derivability of candidate formulas, then scientific validity is improved, but computational complexity increases

Engineering Contradiction:

Improvescientific validityVSAvoidcomputational complexity

Core Design Contradiction:

ReliabilityVSDevice complexity

Solution Approach 1:

The system performs preliminary filtering of candidate formulas using symbolic regression and data-driven methods before submitting them to automated theorem provers. This preliminary action reduces the number of formulas requiring full derivability verification, thereby lowering computational complexity while maintaining scientific validity through multi-stage validation

Inventive Principle:

Principle #10Preliminary action

Solution Approach 2:

The verification process is segmented into multiple stages: initial data fitting, symbolic regression, candidate generation, and final theorem proving. By dividing the verification task into segments, the system manages computational complexity at each stage rather than facing the full complexity burden at once

Inventive Principle:

Principle #1Segmentation

2Loss of information

If symbolic regression is used to discover formulas from data, then interpretability is improved, but accuracy may deteriorate due to limited data

Engineering Contradiction:

ImproveinterpretabilityVSAvoidmodel accuracy

Core Design Contradiction:

Loss of informationVSMeasurement precision

Solution Approach 1:

The system merges symbolic regression with automated theorem proving and background theory integration. This combination allows the system to leverage the interpretability of symbolic regression while compensating for its accuracy limitations through formal verification and theoretical constraints, achieving both interpretability and accuracy

Inventive Principle:

Principle #5Merging (Combining)

Solution Approach 2:

The system uses feedback from multiple sources including data fit quality, theorem proving results, and background theory consistency to iteratively refine candidate formulas. This feedback loop allows the system to improve accuracy by eliminating candidates that fail verification while maintaining interpretability through symbolic representation

Inventive Principle:

Principle #23Feedback

3Measurement precision

If black-box neural networks are used for pattern discovery, then prediction accuracy is improved, but model interpretability deteriorates

Engineering Contradiction:

Improveprediction accuracyVSAvoidmodel interpretability

Core Design Contradiction:

Measurement precisionVSLoss of information

Solution Approach 1:

Instead of using black-box neural networks and then trying to interpret them, the system inverts the approach by directly generating interpretable symbolic formulas through symbolic regression. This inversion prioritizes interpretability from the outset while using theorem proving to ensure accuracy, reversing the conventional neural network paradigm

Inventive Principle:

Principle #13The other way round (Inversion)

4Device complexity

If fixed-form regression is used, then model simplicity is improved, but flexibility in discovering new relationships deteriorates

Engineering Contradiction:

Improvemodel simplicityVSAvoidflexibility

Core Design Contradiction:

Device complexityVSAdaptability or versatility

Solution Approach 1:

The system employs dynamic formula generation through symbolic regression, where the functional form is not fixed in advance but emerges dynamically from the data and background theory. This dynamic approach allows the system to adapt to different phenomena while maintaining relative simplicity through automated generation rather than manual specification

Inventive Principle:

Principle #15Dynamics

Data Source

PatentUS20240330710A1Background theory-based method for refinement and evaluation of functional models extracted from numerical data

Publication Date: 2024.10.03 INTERNATIONAL BUSINESS MACHINE CORPORATION

AI summary

A method generates automated discovery of new scientific formulas. The method includes receiving a background theory associated with a phenomenon being studied. The processor receives a set of training data associated with the phenomenon being studied. The set of training data is processed in a machine learning model that generates candidate formulas from data points in the set of training data. Values of a numerical error-vector are generated for the candidate formulas. The candidate formulas are processed in a reasoning model. The operation of the reasoning model includes generating values of a theoretical error-vector based on the background theory. An output of a performance metric is generated based on a generalization of the theoretical error-vector and a reasoning error. The processor determines whether one of the candidate formulas is a meaningful and valid new scientific formula, based on a behavior of the reasoning error and the reasoning performance metric.