DISCOVER: A Physics-Informed, GPU-Accelerated Symbolic Regression Framework

DISCOVER is an open-source, GPU-accelerated symbolic regression framework designed to discover interpretable, physics-motivated mathematical descriptors by allowing users to incorporate domain knowledge and constrain search spaces within modern Python workflows.

The Gist

The Big Idea: The "Scientific Detective" Tool

Imagine you are a detective trying to solve a mystery. You have a mountain of clues—thousands of measurements like temperature, pressure, weight, and speed—but you don't know how they relate to each other. You want to find the "Golden Rule" (a simple math formula) that explains exactly why a certain material behaves the way it does.

Usually, computers try to solve this by being "black boxes"—they give you an answer, but they can't explain why. It’s like a witness saying, "The thief was wearing red," but being unable to tell you how they know.

DISCOVER is a new piece of software that acts like a Super-Detective. It doesn't just find an answer; it finds a simple, elegant mathematical formula that a human scientist can actually read, understand, and use to predict the future.

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The Three Superpowers of DISCOVER

To make this "detective work" happen, the researchers gave DISCOVER three specific superpowers:

1. The "Common Sense" Filter (Physics-Informed Constraints)

Imagine you are trying to solve a recipe for a cake. If a computer suggests a recipe that calls for "3 gallons of salt" and "2 drops of flour," you’d know it’s nonsense, even if the math technically "works."

In science, math has to follow the laws of physics. You can't add "meters" to "seconds" and get "kilograms"—that’s nonsense. Most AI tools struggle with this, but DISCOVER has "Common Sense." It uses a built-in "unit checker" to instantly throw away any formula that breaks the laws of physics. This saves time by not chasing "impossible" answers.

2. The "Turbo Engine" (GPU Acceleration)

Searching for the perfect formula is like trying to find a specific grain of sand on a massive beach. If you use a spoon (a standard computer processor), it will take years.

DISCOVER uses a "Turbo Engine" (GPU acceleration). Instead of one person looking through the sand with a spoon, it’s like hiring a thousand people with tiny shovels to scan the beach all at once. This allows scientists to search through massive amounts of data in a fraction of the time.

3. The "Minimalist Artist" (Parsimony/Sparsity)

If you ask an artist to draw a dog, you want a clear sketch, not a million tiny dots that eventually look like a dog. In science, a formula with 50 different variables is useless because no human can understand it.

DISCOVER is a "Minimalist." It follows a rule called "Parsimony." It looks for the simplest, shortest, and cleanest formula that still gets the job done. It wants the "elegant sketch," not the "mess of dots."

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Why does this matter?

Scientists use this to discover better materials for the real world. For example:

• Better Batteries: Finding the "Golden Rule" for how ions move inside a battery so we can make electric cars drive further.
• New Medicines: Understanding how molecules interact so we can design better drugs.
• Stronger Materials: Discovering why certain metals stay strong under extreme heat.

In short: DISCOVER helps scientists stop guessing and start discovering the fundamental rules of nature using math that humans can actually understand.

Technical Summary

Technical Summary: DISCOVER Framework

DISCOVER (Data-Informed Symbolic Combination of Operators for Variable Equation Regression) is a new open-source, Python-native symbolic regression (SR) framework designed specifically for scientific disciplines such as computational physics, chemistry, and materials science.

1. The Problem: The Gap in Symbolic Regression

While Symbolic Regression is a powerful tool for discovering interpretable mathematical relationships (descriptors) from data, existing tools face three primary bottlenecks:

• Lack of Physical Integration: Many SR methods are "black-box" and do not easily allow researchers to incorporate a priori domain knowledge (e.g., dimensional consistency or specific functional forms), leading to mathematically accurate but physically nonsensical models.
• Workflow Incompatibility: Existing high-performance tools (like SISSO) often lack seamless integration with modern, Python-based scientific ecosystems.
• Computational Scaling: Exploring large symbolic search spaces is computationally expensive (NP-hard). Many current tools struggle to leverage modern hardware acceleration (GPUs) to handle large-scale, data-intensive studies.

2. Methodology: Guided Search and Optimization

DISCOVER approaches the problem by treating symbolic regression as an $L_0$-regularized least-squares regression problem. The goal is to find a sparse linear combination of candidate features $\Phi$ that minimizes the error relative to a target $y$, subject to a maximum dimensionality constraint $D$:
$$\min_{\beta} \|y - \Phi\beta\|_2^2 \quad \text{subject to} \quad \|\beta\|_0 \leq D$$

To solve this NP-hard problem, the framework employs several key methodological pillars:

• Modular Search Strategies: Instead of a single approach, it offers multiple sparsifying strategies, including heuristic, optimization-based, and stochastic methods such as Orthogonal Matching Pursuit (OMP), Mixed-Integer Quadratic Programming (MIQP), and Simulated Annealing.
• Physics-Informed Constraints: The framework uses a configuration-based interface to allow users to inject domain expertise. A standout feature is the integration with the pint library, which enforces dimensional consistency. This allows the engine to prune dimensionally invalid expressions early in the search process, significantly reducing the search space.
• Hardware-Aware Architecture: To handle the computational load, DISCOVER implements parallelized feature generation and model evaluation. It provides optional acceleration via CUDA (for NVIDIA GPUs) and Metal Performance Shaders (MPS) (for Apple Silicon), while maintaining efficient CPU execution for smaller tasks.

3. Key Contributions

• A Physics-First Design: Unlike general-purpose SR tools that prioritize pure predictive power, DISCOVER prioritizes interpretability and physical consistency by allowing explicit constraints on operators and variable combinations.
• High-Performance Scalability: By incorporating GPU acceleration, the framework bridges the gap between user-friendly Python workflows and the high-performance requirements of large-scale materials discovery.
• User-Centric Configuration: The ability to define constraints through configuration files—rather than modifying source code—makes the tool accessible and reproducible for experimental scientists.

4. Results and Scope

While the provided text is a framework description rather than a results paper, it defines the intended performance outcomes:

• Efficiency: Reduced search space through early pruning of invalid physical dimensions.
• Versatility: Capable of running on everything from local development laptops (via CPU/MPS) to high-performance computing (HPC) clusters (via CUDA).
• Target Applications: The framework is optimized for discovering descriptors in complex fields, such as identifying crystal structure stability or ion mobility in energy storage materials.

5. Significance

DISCOVER represents a shift in symbolic regression from "automated discovery" toward "guided discovery." By empowering researchers to steer the mathematical search using known physical laws, the framework ensures that the resulting models are not just statistically significant, but scientifically meaningful. This makes it a vital tool for the "Materials Genome" era, where the goal is to find simple, interpretable descriptors that can guide the design of new functional materials.