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Project 1 Loss FunctionðŸŠķ

Variationally Motivated Loss Function

Rather than solving the equations of motion exactly, the Euler-Lagrange residual

\[ \mathcal{L}_\text{phys} = \Biggl\langle \bigg( \frac{d}{dt} \frac{\partial {L}}{\partial \dot{q}} - \frac{\partial L}{\partial q} \bigg)^2 \Biggr\rangle \]

is penalized at collocation points in time. Here, \(L\) denotes the Lagrangian of the system. For project 1, the mass and spring constant are normalized so that the above equation can be simplified to

\[ \mathcal{L}_\text{phys} = \bigl\langle (\ddot{q} + \omega^2 q)^2 \bigr\rangle \, . \]

This loss function encourages the system to respect physical dynamics according to the Euler-Lagrange equations.

Note on Low Fidelity

Soft contraints correspond with low fidelity. This is because the physical dynamics we want the model to respect are not directly enforced. Specificlally, the residual of the stationary condition of the action is minimized rather than the action itself.