# Girsanov Drift-Energy Budget

- ID: 862
- Canonical URL: https://synthcore.org/idea/862/girsanov-drift-energy-budget
- API JSON: https://synthcore.org/api/idea/862.json
- API Markdown: https://synthcore.org/api/idea/862.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.13391](https://arxiv.org/abs/2607.13391)
- Category: dynamics
- Solves: stability, generalization, sample-efficiency
- ML areas: ssm, world-model, diffusion, regularization, training-dynamics
- Math tags: probability, stochastic-processes, control-theory, information-theory
- Ratings: usefulness 7/10; difficulty 5/10; novelty 5/10

## Idea description

Regularize a neural continuous-time drift by the quadratic control energy required to move it away from a reference drift. Girsanov’s identity makes this an interpretable path-distribution constraint: expected normalized drift energy equals the relative entropy between controlled and reference trajectory laws.

## Mathematical statement

Consider a reference Ito process dZ_t = b_0(Z_t,t) dt + sigma(Z_t,t) dW_t and a neural controlled process dZ_t = b_theta(Z_t,t) dt + sigma(Z_t,t) dW_t, with an invertible diffusion matrix sigma. Define the normalized drift control u_theta(z,t) = sigma(z,t)^(-1)(b_theta(z,t) - b_0(z,t)). Under standard integrability conditions, Girsanov’s theorem gives a path log-density ratio containing a stochastic integral minus one half of the quadratic energy. Taking expectation removes the martingale and yields D_KL(P_theta || P_0) = one half times E under P_theta of the integral from 0 to T of ||u_theta(Z_t,t)||^2 dt. Therefore an energy budget E <= 2 epsilon predicts path KL no larger than epsilon, up to discretization and measure-change assumptions.

## Key formulas

- $$dZ_t=b_0(Z_t,t)\,dt+\sigma(Z_t,t)\,dW_t,\qquad dZ_t=b_\theta(Z_t,t)\,dt+\sigma(Z_t,t)\,dW_t$$
- $$u_\theta(z,t)=\sigma(z,t)^{-1}\bigl(b_\theta(z,t)-b_0(z,t)\bigr)$$
- $$\log\frac{dP_\theta}{dP_0}=\int_0^T u_\theta(Z_t,t)^\top dW_t-\frac{1}{2}\int_0^T\|u_\theta(Z_t,t)\|^2dt$$
- $$D_{KL}(P_\theta\Vert P_0)=\frac{1}{2}\mathbb{E}_{P_\theta}\int_0^T\left\|\sigma(Z_t,t)^{-1}(b_\theta-b_0)(Z_t,t)\right\|^2dt$$

## Implementation notes

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.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: no_effect
- Verdict: Built and ran a reproducible Girsanov drift-energy MVP for a 1D Ornstein–Uhlenbeck reference SDE with constant controlled drift. Empirical Euler path KL matched the predicted half control energy within 1.01% across tested timesteps; energy divided by c² was constant, and increasing lambda monotonically reduced energy with the analytic optimizer matched to 6e-8. The mechanism is confirmed in this linear-Gaussian toy, but broad neural-SDE superiority was not established.

### Mechanism check

- Verdict: Built and ran a reproducible Girsanov drift-energy MVP for a 1D Ornstein–Uhlenbeck reference SDE with constant controlled drift. Empirical Euler path KL matched the predicted half control energy within 1.01% across tested timesteps; energy divided by c² was constant, and increasing lambda monotonically reduced energy with the analytic optimizer matched to 6e-8. The mechanism is confirmed in this linear-Gaussian toy, but broad neural-SDE superiority was not established.
- Confidence: 8/10
- Limitations: Only a 1D constant-control OU process was tested; no neural drift, nonlinear bistable SDE, hard dual energy budget, multidimensional diffusion solve, long-horizon stability, GPU use, or wall-clock/FLOP comparison was evaluated.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 1
- Paired wins: 0/8
- Benchmark verdict: no measurable effect

## Artifacts

- [bench_girsanov.py](https://synthcore.org/code/348/bench_girsanov.py)
- [bench_report.json](https://synthcore.org/code/348/bench_report.json)
- [girsanov_experiment.py](https://synthcore.org/code/348/girsanov_experiment.py)
- [report.md](https://synthcore.org/code/348/report.md)
- [report_bench_2026-08-31T020641.md](https://synthcore.org/code/348/report_bench_2026-08-31T020641.md)
- [results.json](https://synthcore.org/code/348/results.json)
- [Download all files as ZIP](https://synthcore.org/download/348)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.