SymbolFit: Automatic Parametric Modeling with Symbolic Regression

We introduce SymbolFit, a framework that automates parametric modeling by using symbolic regression to perform a machine-search for functions that fit the data, while simultaneously providing uncertainty estimates in a single run. Traditionally, constructing a parametric model to accurately describe binned data has been a manual and iterative process, requiring an adequate functional form to be determined before the fit can be performed. The main challenge arises when the appropriate functional forms cannot be derived from first principles, especially when there is no underlying true closed-form function for the distribution. In this work, we address this problem by utilizing symbolic regression, a machine learning technique that explores a vast space of candidate functions without needing a predefined functional form, treating the functional form itself as a trainable parameter. Our approach is demonstrated in data analysis applications in high-energy physics experiments at the CERN Large Hadron Collider (LHC). We demonstrate its effectiveness and efficiency using five real proton-proton collision datasets from new physics searches at the LHC, namely the background modeling in resonance searches for high-mass dijet, trijet, paired-dijet, diphoton, and dimuon events. We also validate the framework using several toy datasets with one and more variables.

SymbolNet: Neural Symbolic Regression with Adaptive Dynamic Pruning

Contrary to the use of genetic programming, the neural network approach to symbolic regression can scale well with high input dimension and leverage gradient methods for faster equation searching. Common ways of constraining expression complexity have relied on multistage pruning methods with fine-tuning, but these often lead to significant performance loss. In this work, we propose SymbolNet, a neural network approach to symbolic regression in a novel framework that enables dynamic pruning of model weights, input features, and mathematical operators in a single training, where both training loss and expression complexity are optimized simultaneously. We introduce a sparsity regularization term per pruning type, which can adaptively adjust its own strength and lead to convergence to a target sparsity level. In contrast to most existing symbolic regression methods that cannot efficiently handle datasets with more than $O$(10) inputs, we demonstrate the effectiveness of our model on the LHC jet tagging task (16 inputs), MNIST (784 inputs), and SVHN (3072 inputs).

Symbolic Regression on FPGAs for Fast Machine Learning Inference

The high-energy physics community is investigating the feasibility of deploying machine-learning-based solutions on Field-Programmable Gate Arrays (FPGAs) to improve physics sensitivity while meeting data processing latency limitations. In this contribution, we introduce a novel end-to-end procedure that utilizes a machine learning technique called symbolic regression (SR). It searches equation space to discover algebraic relations approximating a dataset. We use PySR (software for uncovering these expressions based on evolutionary algorithm) and extend the functionality of hls4ml (a package for machine learning inference in FPGAs) to support PySR-generated expressions for resource-constrained production environments. Deep learning models often optimise the top metric by pinning the network size because vast hyperparameter space prevents extensive neural architecture search. Conversely, SR selects a set of models on the Pareto front, which allows for optimising the performance-resource tradeoff directly. By embedding symbolic forms, our implementation can dramatically reduce the computational resources needed to perform critical tasks. We validate our procedure on a physics benchmark: multiclass classification of jets produced in simulated proton-proton collisions at the CERN Large Hadron Collider, and show that we approximate a 3-layer neural network with an inference model that has as low as 5 ns execution time (a reduction by a factor of 13) and over 90% approximation accuracy.