✗ Failed on benchmark 2026

Infinity Atlas for Polynomial Neural Flows

Implementation & benchmark of arXiv:2607.14269 — Blow-up Parameter Landscapes for Polynomial Dynamical Systems

Usefulness8/10

Difficulty6/10

Novelty8/10

Dynamics Stability Accuracy Mlp Rnn Ssm Training dynamics Dynamical systems Bifurcations Numerical analysis

Source paper: Blow-up Parameter Landscapes for Polynomial Dynamical Systems arXiv:2607.14269 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

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.

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)).

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.

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

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Inconclusive

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

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.

Agent confidence: 9/10

Baseline: Optimized lambda=1.2000; diverged from r0=1.25 at t=0.3811; final radius approximately 1.0e4.
Idea: Optimized lambda=0.4919 below the safe boundary 0.62; no divergence over t<=10; final radius=1.18e-4. Boundary predicted/observed: 0.6400/0.6250. Alpha slopes predicted/observed: 1.0/1.0 and 0.5/0.5. Direct numerical blow-up slopes versus lambda and initial radius predicted/observed: -1/-1.000000 and -2/-2.000000.

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.

How to run: python3 infinity_atlas_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.14269) Sep 2, 2026

Ahead of the baseline (−15.5%) but not statistically significant (p=0.1416, wins 5 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0005
Idea mean: 0.0005
Effect (Δ): -0.0001 (−15.5%; negative = idea better)
Wins: 5 / 8 paired seeds
p-value: 0.1416 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±21.6%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 3 configs

Limitations:

Only the built-in dynamics/rnn_small track was tested; no neural ODE, DEQ, continuous-depth residual model, sphere candidate refinement, general leading-term estimator, compactification dynamics, norm-clipping comparator, or larger-scale training was evaluated. The benchmark used 18 epochs and 400 training examples, and the atlas penalty targeted the GRU candidate-state block via the tanh cubic Taylor term rather than an explicitly polynomial neural-flow vector field.

How to run: cd /home/maxwelhelp/all/math2nn/experiments/exp899_infinity-atlas-for-polynomial-neural-flo && /home/maxwelhelp/main/bin/python3 stage2_infinity_atlas_bench.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 30, 2026 latest Failed ✗

Agent confidence: 9/10

Built and evaluated an Infinity Atlas cubic leading-term penalty on the candidate recurrent block of the standard dynamics rnn_small model, using the matched actuated-pendulum task and identical training budgets. The idea achieved lower mean test MSE than the tuned baseline, but the paired delta was not significant (p=0.14155), and the measured trained-model signature contradicted the predicted reduction in positive radial coefficients. Therefore this is a promising but unconfirmed signal, not a demonstrated benchmark win.

Baseline: Tuned lr=0.01; full 8-seed test MSE mean 0.0005451723, std 0.0002036184.
Idea: Best atlas result over lr={0.001,0.003,0.01}; full 8-seed test MSE mean 0.0004606692, std 0.0001829368; paired delta=-0.0000845031, 5/8 wins, permutation p=0.14155.
Math check: Exact cubic radial blow-up scaling predicted log-time slope -1.0 and observed -1.0; angular equilibrium residual 0.0.
Mechanism signature: Baseline mean trained-model max radial cubic coefficient 0.00650875; atlas mean 0.00811023; predicted reduction not confirmed.

Benchmark result