This paper introduces scour physics-informed neural networks (SPINNs), a hybrid physics-data-driven framework for bridge scour prediction using deep learning. SPINNs are developed based on historical scour monitoring data and integrate physics-based empirical equations into neural networks as supplementary loss components. We incorporated three architectures: LSTM, CNN, and NLinear as the base data-driven model. Despite varying performance across different base models and bridges, SPINNs overall outperformed pure data-driven models. In some bridge cases, SPINN reduced forecasting errors by up to 50 percent. In this study, we also explored general models for bridge clusters, trained by aggregating datasets across multiple bridges in a region. The pure data-driven models mostly benefited from this approach, in particular bridges with limited data. However, bridge-specific SPINNs provided more accurate predictions than general SPINNs for almost all case studies. Also, the time-dependent empirical equations derived from SPINNs showed reasonable accuracy in estimating maximum scour depth, providing more accurate predictions compared to HEC-18. Comparing both SPINNs and pure deep learning models with traditional HEC-18 equation indicates substantial improvements in scour prediction accuracy. This study can pave the way for hybrid physics-machine learning methodologies to be implemented for bridge scour design and maintenance.

PySR is an open-source library for practical symbolic regression, a type of machine learning which aims to discover human-interpretable symbolic models. PySR was developed to democratize and popularize symbolic regression for the sciences, and is built on a high-performance distributed back-end, a flexible search algorithm, and interfaces with several deep learning packages. PySR's internal search algorithm is a multi-population evolutionary algorithm, which consists of a unique evolve-simplify-optimize loop, designed for optimization of unknown scalar constants in newly-discovered empirical expressions. PySR's backend is the extremely optimized Julia library SymbolicRegression.jl, which can be used directly from Julia. It is capable of fusing user-defined operators into SIMD kernels at runtime, performing automatic differentiation, and distributing populations of expressions to thousands of cores across a cluster. In describing this software, we also introduce a new benchmark, "EmpiricalBench," to quantify the applicability of symbolic regression algorithms in science. This benchmark measures recovery of historical empirical equations from original and synthetic datasets.