Girsanov Drift-Energy Budget

Apply this to a neural ODE, neural SDE, continuous-depth residual network, or diffusion sampler. Choose b_0 as a pretrained stable drift, a low-capacity baseline, or the drift from the previous training checkpoint, while initially keeping sigma fixed. On each simulated trajectory and time-grid point t_k, evaluate delta = b_theta(z_k,t_k) - b_0(z_k,t_k), solve sigma(z_k,t_k) u_k = delta without explicitly forming an inverse, and accumulate energy = 0.5 times the sum over k of dt times squared norm of u_k. Optimize task_loss + lambda times energy. For a hard budget E_max, use a dual update: loss = task_loss + alpha times (mean_energy - E_max), followed by alpha = max(0, alpha + rho times (mean_energy - E_max)). The paper supplies the Girsanov likelihood ratio and KL-energy identity; trajectory averaging, likelihood-ratio estimation, and numerical integration are empirical. First test a two-dimensional neural SDE on a damped bistable system. Compare unconstrained drift learning, ordinary L2 drift regularization, and Girsanov-energy regularization while sweeping E_max. The predicted signature is estimated path KL approximately equal to estimated energy, with agreement within about 20 percent as the time step is refined. Increasing lambda should reduce both energy and path divergence monotonically. If b_0 is stable, long-horizon trajectory error should increase when the allowed energy budget becomes large enough for b_theta to leave the reference basin.