  Exploring the intersection of deterministic and stochastic dynamics, this
paper delves into Lagrangian discovery for conservative and non-conservative
systems under stochastic excitation. Traditional Lagrangian frameworks, adept
at capturing deterministic behavior, are extended to incorporate stochastic
excitation. The study critically evaluates recent computational methodologies
for learning Lagrangians from observed data, highlighting the limitations in
interpretability and the exclusion of stochastic excitation. To address these
gaps, an automated data-driven framework is proposed for the simultaneous yet
uncoupled discovery of Lagrange densities and the volatility function of
stochastic excitation by leveraging the sparse regression. This novel framework
offers several advantages over existing approaches. Firstly, it provides an
interpretable description of the underlying Lagrange density, allowing for a
deeper understanding of system dynamics under stochastic excitations. Secondly,
it identifies the interpretable form of the generalized stochastic force,
addressing the limitations of existing deterministic approaches. Additionally,
the framework demonstrates robustness and versatility through numerical case
studies encompassing both stochastic differential equations (SDEs) and
stochastic partial differential equations (SPDEs), with results showing almost
exact approximations to true system behavior and minimal relative error in
derived equations of motion.
