# Power-Balanced Modular Neural Block

- ID: 1829
- Canonical URL: https://synthcore.org/idea/1829/power-balanced-modular-neural-block
- API JSON: https://synthcore.org/api/idea/1829.json
- API Markdown: https://synthcore.org/api/idea/1829.md
- Verification status: beats_baseline
- Source: [arXiv:2608.05143](https://arxiv.org/abs/2608.05143)
- Category: architecture
- Solves: stability, generalization, accuracy
- ML areas: mlp, ssm, rnn, training-dynamics
- Math tags: control-theory, dynamical-systems, linear-algebra, geometry
- Ratings: usefulness 8/10; difficulty 5/10; novelty 5/10

## Idea description

Represent each neural module as a Hamiltonian storage system and connect modules through a state-dependent skew or Dirac interconnection instead of arbitrary residual additions. The coupling may change with the hidden state, but its internal power contribution cancels exactly, so total stored energy is controlled only by external inputs and explicitly added dissipation.

## Mathematical statement

Let x be the concatenated states of K neural modules, H(x)=sum_i H_i(x_i) a nonnegative storage function, and e=gradient H(x) the effort vector. A practical port-Hamiltonian neural flow is x_dot = [J(x)-R(x)]e + B(x)u, where J(x) is skew-symmetric, R(x) is positive semidefinite, u is an external feature input, and B(x) is an input map. The skew coupling contributes e^T J(x)e=0 pointwise, even when J depends on x. Therefore H_dot = -e^T R(x)e + u^T B(x)^T e. The underlying Dirac interconnection enforces cancellation of paired internal powers e_c^T f_c and -e_c^T f_c between subsystems.

## Key formulas

- $$D_1 circ D_2 = {((f_1,f_2),(e_1,e_2)) : exists (f_c,e_c), ((f_1,f_c),(e_1,e_c)) in D_1, ((f_2,f_c),(e_2,-e_c)) in D_2}$$
- $$e_1^T f_1 + e_c^T f_c = 0,   e_2^T f_2 - e_c^T f_c = 0  =>  e_1^T f_1 + e_2^T f_2 = 0$$
- $$x_dot = [J(x)-R(x)] gradient_H(x) + B(x)u,   J(x) = -J(x)^T,   R(x) >= 0$$
- $$H_dot = gradient_H(x)^T x_dot = -gradient_H(x)^T R(x) gradient_H(x) + u^T B(x)^T gradient_H(x)$$

## Implementation notes

Use this construction as a replacement for a residual MLP block or as the transition map of a recurrent or state-space layer. Split the hidden vector into K module states x_1 through x_K. Define H(x)=sum_i H_i(x_i), with each H_i implemented as softplus of a small scalar network plus epsilon times the squared norm of x_i; obtain e=gradient H(x) using automatic differentiation. Parameterize the coupling as J(x)=A(x)-A(x)^T, where A(x) is a small low-rank or block-sparse neural network. Parameterize dissipation as R(x)=L(x)L(x)^T+r_min I, with r_min nonnegative. Compute e, then J, L, R, and v=(J-R)e+B(x)u, and update x by Euler, RK4, or a stable semi-implicit integrator. Pseudocode: H=sum(H_i); e=grad(H,x); A=net_A(x); J=A-A.T; L=net_L(x); R=L@L.T+r_min*I; dx=(J-R)@e+B(x)@u; x=x+eta*dx. The paper supplies the power-cancellation construction; positivity of H, matrix parameterizations, and discretization are implementation choices. First experiment: compare this block with an unconstrained residual MLP and a fixed-skew block in a two-module MNIST MLP at matched parameter count. Log H, input power, energy drift, Jacobian singular values, and accuracy. The prediction is that with u=0, continuous-time H decreases whenever R is positive definite. For explicit Euler, the one-step energy defect is O(eta^2); halving eta should reduce the normalized power-balance error by about fourfold. Removing the skew constraint should create an O(eta) energy drift and earlier exploding trajectories.

## Verification

- Status: beats_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: helps
- Verdict: Built and ran a compact power-balanced block experiment with state-dependent skew coupling, positive dissipation, and an input port. The mechanism manifested: skew power cancellation was approximately 1.2e-15, Euler energy-defect scaling had observed slope 2.000, and the measured stability boundary agreed with 2r/(r^2+w^2) to 0.0625%. The PH model also reached much lower synthetic regression MSE, although that comparison was secondary and not parameter-matched.

### Mechanism check

- Verdict: Built and ran a compact power-balanced block experiment with state-dependent skew coupling, positive dissipation, and an input port. The mechanism manifested: skew power cancellation was approximately 1.2e-15, Euler energy-defect scaling had observed slope 2.000, and the measured stability boundary agreed with 2r/(r^2+w^2) to 0.0625%. The PH model also reached much lower synthetic regression MSE, although that comparison was secondary and not parameter-matched.
- Confidence: 9/10
- Limitations: The regression task is synthetic and small, and the baseline was not exactly parameter-matched. The numerical verification uses a quadratic storage function, while the learned block uses diagonal dissipation rather than a full learned nonlinear H_i and dense state-dependent L(x).

### Practical benchmark

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

## Artifacts

- [bench_report.json](https://synthcore.org/code/655/bench_report.json)
- [power_block_experiment.py](https://synthcore.org/code/655/power_block_experiment.py)
- [report.md](https://synthcore.org/code/655/report.md)
- [report_bench_2026-08-31T205841.md](https://synthcore.org/code/655/report_bench_2026-08-31T205841.md)
- [stage2_bench.py](https://synthcore.org/code/655/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/655)

## Disclaimer

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