# Infinity Atlas for Polynomial Neural Flows

- ID: 899
- Canonical URL: https://synthcore.org/idea/899/infinity-atlas-for-polynomial-neural-flows
- API JSON: https://synthcore.org/api/idea/899.json
- API Markdown: https://synthcore.org/api/idea/899.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.14269](https://arxiv.org/abs/2607.14269)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: mlp, rnn, ssm, training-dynamics
- Math tags: dynamical-systems, bifurcations, numerical-analysis
- Ratings: usefulness 8/10; difficulty 6/10; novelty 8/10

## Idea description

For a neural ODE, residual flow, or deep equilibrium model with a dominant polynomial component, compute the directional dynamics induced by its highest-degree homogeneous term on the unit sphere. Penalize or reject parameter regions containing radially growing attracting directions, preventing finite-time activation blow-up while preserving nonlinear dynamics in safe directions.

## Mathematical statement

Let the vector field be F(x; lambda) = F_n(x; lambda) + F_lower(x; lambda), where F_n is homogeneous of degree n greater than 1. Write x = r u, with r = ||x|| and u on the unit sphere. At large r, the radial dynamics are dr/dt = r^n a(u) plus lower-order terms, where a(u) = u transpose F_n(u). The angular dynamics are du/dtau = G(u), where G(u) = F_n(u) - a(u)u and d tau = r^(n-1) dt. An equilibrium direction at infinity satisfies G(u_star) = 0, equivalently F_n(u_star) = alpha_star u_star, with alpha_star = a(u_star). If alpha_star is positive and u_star is attracting for the angular flow, then trajectories approaching that direction blow up in finite time. The compactification z = x divided by sqrt(1 + ||x||^2), y = 1 divided by sqrt(1 + ||x||^2) maps infinity to the boundary y = 0.

## Key formulas

- F(x; lambda) = F_n(x; lambda) + F_lower(x; lambda), with F_n(c x; lambda) = c^n F_n(x; lambda), n > 1.
- dr/dt = r^n a(u) + O(r^(n-1)), and du/dtau = G(u) + O(r^(-1)).
- a(u) = u^T F_n(u; lambda), and G(u) = F_n(u; lambda) - a(u)u.
- G(u_star) = 0 implies F_n(u_star; lambda) = alpha_star u_star; if alpha_star > 0, then r(t) approximately equals [r_0^(1-n) - (n-1) alpha_star (t - t_0)]^(-1/(n-1)).

## Implementation notes

1. Integration point: apply this method to a continuous-depth residual block, neural ODE, polynomial RNN, or DEQ vector field. Explicitly expose the quadratic or cubic part when possible. For a general neural vector field, estimate the leading component by evaluating F(Ru) at several large radii R and fitting the coefficient of R^n. 2. Pseudocode: sample unit vectors u_i; compute v_i = F_n(u_i); compute a_i = dot(u_i, v_i) and g_i = v_i - a_i u_i; add beta times softplus(a_i - margin) when norm(g_i) is below an angular-equilibrium threshold. Periodically refine promising candidates with a constrained optimizer on the sphere minimizing norm(G(u)). For each candidate u_star, estimate the tangent Jacobian using automatic differentiation or finite differences, with P = I - u_star u_star transpose and J_T = P D G(u_star) P. Reject or damp updates when alpha_star is positive and all relevant tangent eigenvalues have negative real parts. 3. Computed from the paper: compactification, leading-degree reduction, directional equilibrium equation, and radial blow-up law. Estimated empirically: sphere coverage, candidate equilibria, and tangent eigenvalues. 4. First cheap experiment: train a two-dimensional cubic neural ODE on a spiraling trajectory dataset, comparing unconstrained training, norm clipping, and the atlas penalty. Sweep the cubic coefficient lambda and integrate from many initial states. Prediction: the first finite-time numerical divergence occurs within 20 percent of the atlas boundary where an attracting u_star changes from alpha_star less than or equal to zero to alpha_star greater than zero. In the blow-up regime, log(r) versus log(t_star - t) has slope approximately -1/(n-1), equal to -1/2 for a cubic field.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: inconclusive
- Verdict: Built a 2D polynomial-flow Infinity Atlas verifier with cubic homogeneous dynamics, angular equilibrium and tangent-stability analysis, blow-up detection, direct ODE checks, and a toy baseline-versus-atlas optimization. The predicted instability boundary was lambda*=0.64 versus an observed sweep midpoint of 0.625, a 2.34% error; alpha scaling slopes and direct blow-up-time exponents matched predictions to numerical precision. The atlas penalty prevented divergence in the toy optimization while the unconstrained baseline diverged, although this is not evidence of general neural-network training gains.

### Mechanism check

- Verdict: Built a 2D polynomial-flow Infinity Atlas verifier with cubic homogeneous dynamics, angular equilibrium and tangent-stability analysis, blow-up detection, direct ODE checks, and a toy baseline-versus-atlas optimization. The predicted instability boundary was lambda*=0.64 versus an observed sweep midpoint of 0.625, a 2.34% error; alpha scaling slopes and direct blow-up-time exponents matched predictions to numerical precision. The atlas penalty prevented divergence in the toy optimization while the unconstrained baseline diverged, although this is not evidence of general neural-network training gains.
- Confidence: 9/10
- Limitations: The experiment uses an analytically exposed 2D cubic field rather than a trained neural ODE, residual network, DEQ, or polynomial RNN. The atlas penalty is a hand-tuned toy optimization term and does not include sphere candidate refinement, general leading-term estimation, compactification dynamics, norm clipping comparison, or statistical repeats.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.14155
- Paired wins: 5/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_report.json](https://synthcore.org/code/163/bench_report.json)
- [infinity_atlas_experiment.py](https://synthcore.org/code/163/infinity_atlas_experiment.py)
- [report.md](https://synthcore.org/code/163/report.md)
- [report_bench_2026-08-30T182136.md](https://synthcore.org/code/163/report_bench_2026-08-30T182136.md)
- [results.json](https://synthcore.org/code/163/results.json)
- [stage2_infinity_atlas_bench.py](https://synthcore.org/code/163/stage2_infinity_atlas_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/163)

## Disclaimer

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