# Chemotactic Feature Transport Layer

- ID: 1756
- Canonical URL: https://synthcore.org/idea/1756/chemotactic-feature-transport-layer
- API JSON: https://synthcore.org/api/idea/1756.json
- API Markdown: https://synthcore.org/api/idea/1756.md
- Verification status: failed_benchmark
- Source: [arXiv:2608.03685](https://arxiv.org/abs/2608.03685)
- Category: architecture
- Solves: accuracy, stability
- ML areas: cnn, training-dynamics
- Math tags: dynamical-systems, pde, spectral-theory
- Ratings: usefulness 7/10; difficulty 6/10; novelty 8/10

## Idea description

Replace part of a CNN or continuous-depth feature block with two coupled feature fields. One field is transported up gradients of the other through a conservative cross-gradient flux, creating adaptive spatial organization that ordinary diffusion or symmetric convolution cannot produce. The coupling strength and dominant wavelength are controlled by a directly testable linear-instability boundary.

## Mathematical statement

The model uses two fields u and v, diffusivities D_u and D_v, chemotactic strength beta, and reaction functions f and g. Around a homogeneous state (u_bar, v_bar), let f_u, f_v, g_u, and g_v denote the four partial derivatives of the reaction functions at that state. For a Fourier mode with squared wave number q equal to |k|^2, the linearized matrix is M(q) = [[f_u - D_u q, f_v + beta u_bar q], [g_u, g_v - D_v q]]. The chemotactic term changes an off-diagonal entry by beta u_bar q, unlike diffusion. If the trace is negative, instability occurs when the determinant becomes negative. When g_u u_bar q is positive, the critical chemotactic strength is beta_c(q) = ((f_u - D_u q)(g_v - D_v q) - g_u f_v) / (g_u u_bar q).

## Key formulas

- $$\partial_t u=D_u\Delta u-\beta\nabla\cdot(u\nabla v)+f(u,v),\qquad \partial_t v=D_v\Delta v+g(u,v).$$
- $$M(q)=\begin{pmatrix}f_u-D_uq&f_v+\beta\bar{u}q\\g_u&g_v-D_vq\end{pmatrix},\qquad q=|k|^2.$$
- $$\lambda_{\pm}(q)=\frac{\operatorname{tr}M(q)\pm\sqrt{(\operatorname{tr}M(q))^2-4\det M(q)}}{2},\qquad \text{instability if }\max_q\operatorname{Re}\lambda_+(q)>0.$$
- $$\beta_c(q)=\frac{(f_u-D_uq)(g_v-D_vq)-g_uf_v}{g_u\bar{u}q},\qquad \det M(q)<0\text{ for }\beta>\beta_c(q)\text{ when }g_u\bar{u}q>0.$$

## Implementation notes

Integrate this as a residual feature block in a CNN, U-Net, or continuous-depth vision model. Split a feature tensor into two channel groups U and V, optionally applying softplus so that U is nonnegative. For a step size h, use U_new = U + h times [D_u Delta_h U - beta div_h(U times grad_h V) + F_theta(U,V)] and V_new = V + h times [D_v Delta_h V + G_theta(U,V)]. Compute the chemotactic flux on cell faces using a face-averaged U multiplied by the centered spatial gradient of V, then apply a discrete divergence. This conservative discretization reduces artificial mass creation. Estimate f_u, f_v, g_u, and g_v by automatic differentiation of the small reaction networks F_theta and G_theta at the minibatch mean feature state. For each FFT frequency, construct M(q) and record its largest real eigenvalue. The paper supplies the linearized matrix and instability criterion; feature Jacobians and operating states are estimated empirically. First test CIFAR-10 with a small ResNet, comparing a standard residual block, diffusion-only coupling, and chemotactic coupling at equal parameter count. The prediction is a sharp onset near the smallest measured beta_c(q), and the growth rate of a sinusoidal perturbation should agree with the predicted eigenvalue within 20 percent.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: no_effect
- Verdict: Built a conservative periodic chemotactic transport module with the stated two-field linearization, RK4 perturbation integrator, beta-critical calculation, and finite-volume face flux/divergence. The predicted instability boundary was beta_c=0.5739 at mode q=9; the sweep first became positive at beta=0.6614 on its coarse grid and was exactly zero at beta_c. At beta=0.8239, measured log-amplitude growth was 0.3677804427677 versus predicted eigenvalue 0.3677804427682 (relative error 1.5e-12); mass error of the conservative transport discretization was 0. The mechanism manifested clearly, but this is not evidence of improved image accuracy.

### Mechanism check

- Verdict: Built a conservative periodic chemotactic transport module with the stated two-field linearization, RK4 perturbation integrator, beta-critical calculation, and finite-volume face flux/divergence. The predicted instability boundary was beta_c=0.5739 at mode q=9; the sweep first became positive at beta=0.6614 on its coarse grid and was exactly zero at beta_c. At beta=0.8239, measured log-amplitude growth was 0.3677804427677 versus predicted eigenvalue 0.3677804427682 (relative error 1.5e-12); mass error of the conservative transport discretization was 0. The mechanism manifested clearly, but this is not evidence of improved image accuracy.
- Confidence: 9/10
- Limitations: No CIFAR-10, CNN/ResNet training, learned reaction networks, equal-parameter accuracy comparison, GPU benchmark, or long-horizon nonlinear pattern-spectrum analysis was tested. The onset sweep used a coarse beta grid, so its observed threshold is reported as a bracketed grid observation rather than a refined estimate.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.28255
- Paired wins: 2/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_report.json](https://synthcore.org/code/645/bench_report.json)
- [chemotactic_transport.py](https://synthcore.org/code/645/chemotactic_transport.py)
- [report.md](https://synthcore.org/code/645/report.md)
- [report_bench_2026-08-31T203448.md](https://synthcore.org/code/645/report_bench_2026-08-31T203448.md)
- [results.json](https://synthcore.org/code/645/results.json)
- [stage2_bench.py](https://synthcore.org/code/645/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/645)

## Disclaimer

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