# Front-Calibrated Bistable Neural Field

- ID: 1832
- Canonical URL: https://synthcore.org/idea/1832/front-calibrated-bistable-neural-field
- API JSON: https://synthcore.org/api/idea/1832.json
- API Markdown: https://synthcore.org/api/idea/1832.md
- Verification status: mechanism_failed
- Source: [arXiv:2608.05251](https://arxiv.org/abs/2608.05251)
- Category: architecture
- Solves: stability, accuracy
- ML areas: rnn, ssm, cnn, training-dynamics
- Math tags: dynamical-systems, pde, stability, linear-algebra
- Ratings: usefulness 8/10; difficulty 6/10; novelty 7/10

## Idea description

Construct a spatial recurrent network whose local vector hidden state has two stable attractors and whose neighbor coupling is diffusive. Train or constrain the network so that the desired attractor invades the undesired one with a controlled positive front velocity, rather than relying on a scalar class-frequency variable that can erase depletion and interface structure.

## Mathematical statement

Use the reaction-diffusion neural field \(\partial_t\mathbf{h}=D\Delta\mathbf{h}+\mathbf{f}_{\theta}(\mathbf{h};\mathbf{x})+\sqrt{2TG(\mathbf{h})}\boldsymbol{\eta}\), where \(\mathbf{h}(t,\mathbf{x})\in\mathbb{R}^{k}\) is the vector hidden state, \(D\geq0\) is spatial coupling, \(\mathbf{f}_{\theta}\) is a learned local vector field, \(T\) is noise strength, and \(G\) is a positive semidefinite noise covariance. Require two fixed points \(\mathbf{h}_{+}^{*}\) and \(\mathbf{h}_{-}^{*}\) satisfying \(\mathbf{f}_{\theta}(\mathbf{h}_{\pm}^{*})=0\), with every eigenvalue of the local Jacobian \(J_{\pm}=\partial\mathbf{f}_{\theta}/\partial\mathbf{h}|_{\mathbf{h}_{\pm}^{*}}\) having negative real part. For a planar interface, define \(v\) as the measured interface velocity. The non-equilibrium nucleation approximation uses \(v\) as the bulk driving term and \(\kappa\) as an effective interfacial cost, giving \(\Phi(R)=S_d\kappa R^{d-1}-V_dvR^d\), where \(R\) is droplet radius, \(V_d\) is unit-ball volume, and \(S_d=dV_d\). The critical radius is \(R_c=(d-1)\kappa/v\); droplets larger than \(R_c\) grow and smaller droplets shrink.

## Key formulas

- $$\frac{\partial\mathbf{h}}{\partial t}=D\Delta\mathbf{h}+\mathbf{f}_{\theta}(\mathbf{h};\mathbf{x})+\sqrt{2T G(\mathbf{h})}\,\boldsymbol{\eta}(t,\mathbf{x})$$
- $$\mathbf{f}_{\theta}(\mathbf{h}_{\pm}^{*})=0,\qquad \operatorname{Re}\lambda\!\left(\frac{\partial\mathbf{f}_{\theta}}{\partial\mathbf{h}}(\mathbf{h}_{\pm}^{*})\right)<0$$
- $$\Phi(R)=S_d\kappa R^{d-1}-V_dvR^d,\qquad R_c=\frac{(d-1)\kappa}{v}$$
- $$\Phi(R_c)=\frac{S_d}{d}\kappa\left(\frac{(d-1)\kappa}{v}\right)^{d-1}$$

## Implementation notes

Use a 2-D convolutional recurrent model or neural cellular automaton. Each cell stores a k-dimensional hidden vector and applies an explicit Euler update: h[t+1]=h[t]+dt*(f_theta(h[t],x)+D*Laplacian(h[t])) plus optional noise. Parameterize f_theta as a small MLP and learn two attractor prototypes h_plus and h_minus. Pretrain the local MLP on two synthetic clusters so both prototypes are fixed points. Estimate the Jacobian at each prototype using automatic differentiation and reject or penalize updates when any Jacobian eigenvalue has positive real part. Calibrate the invasion dynamics by initializing a half-plane of h_plus against a half-plane of h_minus, evolving the system, and fitting front velocity v from the slope of interface midpoint versus time. Estimate kappa by initializing circular droplets of several radii, labeling each as growing or shrinking, and fitting the 50-percent transition radius R_c; then set kappa=v*R_c/(d-1). Add a regularizer (v-v_target)^2 and a penalty for unstable local Jacobians. The first cheap experiment is a 32x32 neural cellular automaton on synthetic binary-image denoising or segmentation, compared with a scalar-state cellular automaton and an unconstrained ConvGRU. The predicted signature is a sharp growth transition: droplets with R<R_c contract and droplets with R>R_c expand, with fitted R_c agreeing with (d-1)kappa/v within 20 percent. The vector model should also preserve interface structure and improve long-horizon denoising stability.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a readable 2-D bistable reaction-diffusion neural-field MVP with a vector hidden state, local Jacobian stability checks, front tracking, droplet tests, and a scalar denoising baseline. Two mechanisms manifested: bistability disappeared at the predicted saddle-node boundary |b|=2/(3√3)≈0.3849, and front velocity changed sign at b=0 with approximately linear positive-bias scaling (measured slope 1.93). The nucleation prediction was not confirmed within the requested 20% tolerance: inverse-velocity critical-radius ratios were 0.65, 1.00, and 0.81 across the bias sweep, and denoising accuracy showed no improvement over the scalar baseline.

### Mechanism check

- Verdict: Built a readable 2-D bistable reaction-diffusion neural-field MVP with a vector hidden state, local Jacobian stability checks, front tracking, droplet tests, and a scalar denoising baseline. Two mechanisms manifested: bistability disappeared at the predicted saddle-node boundary |b|=2/(3√3)≈0.3849, and front velocity changed sign at b=0 with approximately linear positive-bias scaling (measured slope 1.93). The nucleation prediction was not confirmed within the requested 20% tolerance: inverse-velocity critical-radius ratios were 0.65, 1.00, and 0.81 across the bias sweep, and denoising accuracy showed no improvement over the scalar baseline.
- Confidence: 9/10
- Limitations: This is an untrained analytic toy rather than a learned MLP or ConvGRU; the auxiliary vector coordinate is locally coupled but does not affect spatial diffusion, noise was not tested, front and droplet measurements use finite grids and coarse integer radii, and no FLOP-matched neural-network comparison was performed.

## Artifacts

- [bistable_field.py](https://synthcore.org/code/656/bistable_field.py)
- [report.md](https://synthcore.org/code/656/report.md)
- [results.json](https://synthcore.org/code/656/results.json)
- [Download all files as ZIP](https://synthcore.org/download/656)

## Disclaimer

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