🤖 machine learning

Towards symbolic regression for interpretable clinical decision scores

This paper introduces Brush, a novel symbolic regression algorithm that integrates decision-tree-like splitting with non-linear constant optimization to generate accurate, interpretable, and rule-based clinical decision scores that outperform or match existing methods while maintaining model simplicity.

Original authors: Guilherme Seidyo Imai Aldeia, Joseph D. Romano, Fabricio Olivetti de Franca, Daniel S. Herman, William G. La Cava

Published 2026-04-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Guilherme Seidyo Imai Aldeia, Joseph D. Romano, Fabricio Olivetti de Franca, Daniel S. Herman, William G. La Cava

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a doctor trying to decide if a patient needs to go to the intensive care unit or if they can go home. You have a massive pile of data: their heart rate, blood pressure, age, temperature, and how many times they've been to the hospital before.

Currently, doctors use "scorecards" (like the CART or MEWS scores). These are like simple flowcharts: "If the heart rate is over 140, add 5 points. If the age is over 70, add 3 points." If the total is high, the patient is at risk. These are great because they are easy to understand and follow.

However, modern computer algorithms (Machine Learning) are like super-smart but mysterious wizards. They can look at all that data and predict risk with incredible accuracy, but they do it in a "black box." You can't ask them why they made a decision, and their logic is often a tangled mess of millions of numbers that no human can read.

The Problem:
We want the accuracy of the wizard but the clarity of the scorecard.
Traditional "Symbolic Regression" (a type of AI that tries to write math formulas) is great at finding simple equations (like $y = 2x + 5$ ), but it struggles to handle the "If/Then" logic that doctors actually use. It's like trying to write a recipe using only a single long sentence without any commas or periods.

The Solution: Meet "Brush"
The authors of this paper created a new AI tool called Brush. Think of Brush as a smart gardener who is trying to grow the perfect, simplest tree to solve a problem.

Here is how Brush works, using a few analogies:

1. The "Split" Branch (The Fork in the Road)

Most math formulas are smooth and continuous. Brush is special because it can grow branches that act like decision points.

Analogy: Imagine you are navigating a maze. A normal math formula is like a straight line. Brush can say, "If you hit a wall (Condition), turn left and follow this path. If not, turn right and follow that path."
This allows Brush to create models that look like the flowcharts doctors use, but it discovers them automatically from the data.

2. The "Tuning Knob" (Fine-Tuning)

Once Brush builds a tree with these branches, it doesn't just leave it there. It has a special tuning knob (non-linear optimization).

Analogy: Imagine you built a radio. You have the antenna and the speakers (the structure), but the sound is fuzzy. Brush has a mechanic that goes in and tightens every screw and adjusts every frequency until the sound is crystal clear.
This ensures the model isn't just "okay," but highly accurate, even when the data is messy or noisy (like a patient with slightly weird vitals).

3. The "Pruning Shears" (Simplification)

Brush is obsessed with simplicity. It uses a technique called inexact simplification.

Analogy: Imagine you wrote a story that was 500 pages long. Brush is an editor who reads it and says, "You don't need this whole paragraph; this one sentence says the same thing." It cuts away the fluff, leaving you with a short, punchy story that still tells the whole truth.
This prevents the model from becoming too complex and hard to understand.

What Did They Find?

The researchers tested Brush in two ways:

The Physics Test (The "Textbook" Challenge): They asked Brush to rediscover famous physics equations (like those from Feynman) from noisy data.
- Result: Brush was a champion. It found the correct equations more often than other AI methods, and the equations it found were much shorter and cleaner. It was like finding the shortest, most elegant path through a forest.
The Hospital Test (The "Real World" Challenge): They fed Brush data from 10,000 emergency room patients to see if it could recreate the doctors' scorecards (CART and MEWS) and predict who was in danger.
- Result: Brush did an amazing job. It learned the rules that doctors use (e.g., "High heart rate + Low blood pressure = Danger") but did it automatically.
- The Magic: It was just as accurate as the "black box" wizards (like Random Forests) but produced models that were small enough to fit on a single page of paper. A doctor could actually read the model and say, "Ah, I see. It's checking the heart rate first, then the blood pressure. That makes sense."

Why Does This Matter?

In healthcare, trust is everything. If an AI tells a doctor to perform surgery, the doctor needs to know why.

Old AI: "I say surgery because my internal math says so." (Doctor: "I don't trust that.")
Brush: "I say surgery because the patient's heart rate is high AND their blood pressure is low, which matches the pattern of a heart attack." (Doctor: "I understand. Let's do it.")

In summary: Brush is a new kind of AI that combines the best of two worlds: the predictive power of complex computers and the clear, logical rules of human decision-making. It helps us build medical tools that are not only smart but also transparent and trustworthy.

1. Problem Statement

Medical decision-making often relies on algorithms that combine risk equations with rule-based logic (e.g., clinical scoring systems like CART or MEWS) to provide standardized treatment pathways. While Symbolic Regression (SR) is powerful for discovering interpretable, data-driven mathematical expressions, traditional SR algorithms face two main limitations in this context:

Lack of Split-Wise Operations: Traditional SR is constrained to continuous function forms, making it difficult to model the discrete, rule-based logic (if-then-else) inherent in clinical triage scores.
Parameter Optimization: Existing SR methods often struggle to optimize non-linear constants effectively, leading to suboptimal convergence or reliance on random constants.

The paper addresses the need for a hybrid approach that can seamlessly integrate decision-tree-like splitting logic with non-linear parameter optimization to create models that are both highly accurate and intrinsically interpretable for clinical decision support.

2. Methodology: The "Brush" Algorithm

The authors propose Brush, a Multi-Objective Genetic Programming (GP) algorithm designed to bridge the gap between symbolic prediction models and split-wise operations.

Core Innovations

Specialized Split Node: Brush introduces a "split node" operator inspired by decision trees. Unlike standard decision trees where splits are fixed at the root or specific layers, Brush allows split nodes to appear anywhere within the expression tree.
- Structure: A split node has three branches: a condition subtree ( $f_c$ ), a true branch ( $f_T$ ), and a false branch ( $f_F$ ).
- Optimization: The algorithm exhaustively tests feature/threshold combinations to find a threshold $\tau^*$ that minimizes the sum of target variances in the resulting partitions. This effectively performs one-dimensional clustering within the symbolic expression.
Hybrid Optimization Strategy: Brush combines evolutionary search with local non-linear optimization (Levenberg-Marquardt).
- Heuristic Process: The optimization occurs in three steps: (1) Optimize the conditional subtree; (2) Find the optimal threshold $\tau^*$ ; (3) Fit the remaining parameters in the true/false branches using the filtered data subsets.
- Weighted Nodes: The algorithm assigns innate weights to nodes (including terminals) which can be toggled on/off, extending mechanisms from previous work (Operon) to allow for more flexible parameter tuning.
Multi-Objective Fitness: Brush optimizes a vector of objectives:
- Performance: Mean Squared Error (MSE) for regression or classification metrics.
- Complexity: Linear complexity (sum of node complexities) to enforce model simplicity and interpretability.
Selection and Survival: It utilizes $\epsilon$ -lexicase selection and NSGA-II (Non-dominated Sorting Genetic Algorithm II) for survival to ensure robust convergence and Pareto-optimal solutions.
Post-Processing: An inexact simplification step replaces large sub-trees with simpler, approximately equivalent alternatives to further compress models.

Classification Extension

For classification tasks, Brush fixes a logistic regression node at the root of the expression tree. This allows the model to output probability estimates while maintaining the interpretability of the underlying symbolic structure.

3. Key Contributions

Algorithmic Novelty: The integration of split-wise operations with non-linear parameter optimization within a single GP framework, allowing for "seamless" rule-based logic in symbolic expressions.
Benchmark Performance: Demonstration of Pareto-optimal performance on the SRBench benchmark (122 real-world datasets and 130 physics equations), outperforming or matching state-of-the-art (SotA) methods like AIFeynman, Operon, and PS-Tree in accuracy while producing significantly smaller models.
Clinical Application: Successful application to Electronic Health Record (EHR) data to recapitulate and improve upon widely used clinical scoring systems (CART and MEWS), generating models that are both accurate and clinically interpretable.

4. Experimental Results

A. Symbolic Regression Benchmark (SRBench)

Black-Box Problems: Brush achieved the second-best rank in test $R^2$ scores while producing models significantly smaller than PS-Tree and often smaller than Operon. It dominated FEAT in both simplicity and accuracy.
Ground-Truth Problems: Brush recovered accurate expressions for physics equations (Feynman and Strogatz datasets) even under high noise levels.
- Noise Robustness: Over 50% of solutions achieved $R^2 > 0.999$ across all noise levels.
- Comparison: Unlike methods like AIFeynman and MRGP, which suffered accuracy drops with increased noise, Brush remained robust.
- Note: PS-Tree was excluded from ground-truth recovery because its enforced split-wise structure prevents exact equation recovery.

B. Clinical Decision Support (MIMIC-IV-ED Dataset)

The authors tested Brush on 10,000 patient records to predict:

Regression: Mean Arterial Pressure (MAP), CART score, and simplified MEWS score.
Classification: High risk of catastrophic deterioration based on the above scores.

Key Findings:

Regression Performance: Brush achieved $R^2 \approx 1.00$ for MAP and CART, and $R^2 \approx 0.74$ for MEWS. Crucially, Brush produced much smaller models (e.g., ~76 nodes for CART) compared to Random Forests (millions of nodes) and Decision Trees (hundreds of nodes).
Classification Performance:
- Brush achieved an AUPRC of 0.99 for CART deterioration and 0.95 for MEWS deterioration.
- Split Nodes are Critical: Experiments disabling split nodes showed a statistically significant drop in classification performance (AUPRC dropped from 0.99 to 0.78 for CART), proving that the split logic is essential for capturing clinical risk patterns.
Interpretability: The generated models mirrored clinical heuristics. For example, the Brush model for CART deterioration prioritized respiratory rate, followed by diastolic blood pressure and heart rate, exactly matching the logic of the original clinical scoring system but in a compact, rule-based form.

5. Significance and Conclusion

The paper demonstrates that Brush effectively balances the trade-off between predictive performance, model complexity, and interpretability.

Clinical Impact: It provides a method to automatically derive "clinical decision scores" from big data (EHRs) that are as accurate as complex "black-box" models (like Random Forests) but as interpretable as manual scoring systems. This is vital for regulatory compliance and clinician trust in AI-driven healthcare.
Scientific Contribution: By enabling split-wise operations within symbolic regression, Brush expands the applicability of SR to domains requiring discrete logic, moving beyond purely continuous function approximation.
Future Work: The authors note that while Brush is promising, it was not fully fine-tuned, and the complexity metrics used are currently arbitrary. Future iterations could optimize these hyperparameters for specific domains.

In summary, Brush represents a significant step forward in Explainable AI (XAI) for healthcare, offering a tool that can learn complex, non-linear, rule-based relationships from data while producing models that clinicians can understand and trust.