Based on the detailed documentation provided for EmpiricalDynamics, the addition of strategic diagrams would significantly enhance user understanding, particularly for the complex workflows involved in SDE discovery and the hybrid architecture.

Here is the revised documentation with suggested image tags inserted at high-value locations.

EmpiricalDynamics

High-Performance & Robust Empirical Discovery of Differential Equations from Time Series Data

EmpiricalDynamics is a comprehensive toolkit for discovering differential and difference equations from empirical time series data. It combines the statistical power of R with a high-performance Julia backend (via SymbolicRegression.jl) to offer a robust engine capable of recovering physical laws, economic models, and stochastic differential equations from noisy data.

Performance Benchmarks:

• Deterministic ODEs: R² > 0.93 on chaotic systems (Lorenz attractor)
• Stochastic SDEs: Drift R² > 0.86, Diffusion R² > 0.59 with GLS refinement
• Physics Constants: Precision of 10⁻⁸ recovering π and e from noisy data