# Dirac-Coupled Energy-Shaping Optimizer

- ID: 1830
- Canonical URL: https://synthcore.org/idea/1830/dirac-coupled-energy-shaping-optimizer
- API JSON: https://synthcore.org/api/idea/1830.json
- API Markdown: https://synthcore.org/api/idea/1830.md
- Verification status: failed_benchmark
- Source: [arXiv:2608.05143](https://arxiv.org/abs/2608.05143)
- Category: optimizer
- Solves: stability, speedup
- ML areas: optimizer, training-dynamics, scheduler
- Math tags: control-theory, dynamical-systems, optimization, linear-algebra, stochastic-processes
- Ratings: usefulness 7/10; difficulty 6/10; novelty 6/10

## Idea description

Construct optimizer variables as interconnected Hamiltonian subsystems: parameters store potential energy, momentum stores kinetic energy, and a skew coupling transfers energy between them without net creation. Positive-semidefinite resistance removes energy and provides an explicit damping knob, separating conservative exploration from dissipative convergence.

## Mathematical statement

Let theta be the model parameters, p the optimizer momentum, z=(theta,p), and H(theta,p)=L(theta)+0.5 p^T M^{-1}p, where L is the training loss and M is a positive-definite mass matrix. Define z_dot=[J(z)-R(z)] gradient_H(z)+B xi_t, where J(z) is skew-symmetric, R(z) is positive semidefinite, and xi_t represents the stochastic minibatch gradient as an external port. In the deterministic autonomous case, H_dot=-gradient_H^T R gradient_H <= 0. With minibatch forcing, the injected power is xi_t^T B^T gradient_H. A practical diagnostic is the energy drift after subtracting this injected power; it should be nonpositive up to time-discretization error.

## Key formulas

- $$H(theta,p) = L(theta) + 0.5 p^T M^(-1) p,   gradient_H = [gradient_theta L(theta); M^(-1)p]$$
- $$z_dot = [J(z)-R(z)] gradient_H(z) + B xi_t,   J(z)=-J(z)^T,   R(z)>=0$$
- $$dH/dt = -gradient_H(z)^T R(z) gradient_H(z) + xi_t^T B^T gradient_H(z)$$
- $$q_hat_t = [H_(t+1)-H_t]/eta - xi_t^T B^T gradient_H_t,   q_hat_t <= O(eta)$$

## Implementation notes

Integrate this mechanism into the optimizer rather than the network forward pass. Maintain theta and p for every parameter tensor. Begin with the canonical interconnection J0=[[0,I],[-I,0]] and optionally add a learned low-rank skew term J(z)=J0+A(z)-A(z)^T. Use R=diag(r_theta I,r_p I), or start with R=diag(0,gamma I) so momentum is damped while position and momentum exchange energy. At minibatch t, compute the batch gradient g_t, set the external forcing xi_t=(g_t,0), evaluate e_t=gradient_H(theta_t,p_t), and update z by z <- z+eta*((J-R)e_t+B xi_t). For improved numerical behavior, integrate the skew Hamiltonian part with symplectic Euler or leapfrog and apply damping implicitly. Pseudocode: g=backprop(batch); H=loss+0.5*p.T@inv(M)@p; e=grad(H,[theta,p]); A=skew_net(theta,p); J=J0+A-A.T; R=positive_damping(); z=z+eta*((J-R)@e+B@concat(g,zeros_like(p))). The paper supplies the interconnection and exact internal power cancellation; minibatch noise statistics and discretization effects must be measured. First experiment: train a small ResNet or three-layer MLP on CIFAR-10 and compare against SGD, momentum SGD, and Adam over identical step-size grids. Log H, gradient norm, momentum norm, injected batch power, and q_hat. The predicted signature is that after subtracting injected power, mean energy drift is nonpositive up to discretization error. For a first-order implementation, the residual should scale approximately linearly with eta; doubling eta should approximately double the residual before instability. An unconstrained J should show positive systematic energy drift and a lower maximum stable learning rate.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: harms
- Verdict: Built a PyTorch Dirac-coupled optimizer with per-parameter momentum, canonical skew exchange, damping, and a quadratic verification suite. The predicted symplectic stability boundary matched six settings within 0.02%, and the curvature scaling sqrt(lambda)*eta_max was 1.9996 versus the predicted 2.0. However, the nonlinear MLP benchmark showed no practical win: Dirac reached loss 0.1149 and 94.25% validation accuracy, versus momentum SGD at loss 0.0863 and 98.50%.

### Mechanism check

- Verdict: Built a PyTorch Dirac-coupled optimizer with per-parameter momentum, canonical skew exchange, damping, and a quadratic verification suite. The predicted symplectic stability boundary matched six settings within 0.02%, and the curvature scaling sqrt(lambda)*eta_max was 1.9996 versus the predicted 2.0. However, the nonlinear MLP benchmark showed no practical win: Dirac reached loss 0.1149 and 94.25% validation accuracy, versus momentum SGD at loss 0.0863 and 98.50%.
- Confidence: 8/10
- Limitations: Only a one-dimensional quadratic and a small synthetic two-dimensional MLP were tested. CIFAR-10, larger networks, stochastic injected-power subtraction, hyperparameter grids, wall-clock/FLOP analysis, learned low-rank skew terms, and multiple random seeds were not evaluated.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_dirac.py](https://synthcore.org/code/681/bench_dirac.py)
- [bench_report.json](https://synthcore.org/code/681/bench_report.json)
- [dirac_optimizer.py](https://synthcore.org/code/681/dirac_optimizer.py)
- [mini_experiment.py](https://synthcore.org/code/681/mini_experiment.py)
- [mini_results.json](https://synthcore.org/code/681/mini_results.json)
- [report.md](https://synthcore.org/code/681/report.md)
- [report_bench_2026-08-31T214357.md](https://synthcore.org/code/681/report_bench_2026-08-31T214357.md)
- [results.json](https://synthcore.org/code/681/results.json)
- [verify_dirac_optimizer.py](https://synthcore.org/code/681/verify_dirac_optimizer.py)
- [Download all files as ZIP](https://synthcore.org/download/681)

## Disclaimer

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