Generator Discovery Lab

Pick the right coordinate chart and a curved law becomes a straight line. Synthetic exponential, power-law and logistic generators are ranked by chart below.

3/3

laws recovered

Synthetic exponential, power-law, and logistic generators recover their expected charts.

2->1

logistic dynamics

The raw SINDy library needs two terms; the logit coordinate needs one constant term.

100%

wrapped validity

Same validity test on both arms: raw grid values pass 13.9%, exp/sigmoid/softmax-mapped values pass 100.0%.

5.4e-16

reaction residual

Fitted log-linear residual on synthetic noiseless mass-action data (exponents are fitted, not set).

New demo trackSynthetic data, computed at build

Every number on this page is computed by lib/generatorDemos.ts from synthetic, noiseless data when the page is built. They are illustrative checks of the method, not experimental results. For live browser solvers, see the molecular, radiation, turbulence, and crack-tip demos.

This page reframes the demos around a testable claim: choose the coordinate chart that makes a model simple, then search inside latent maps that preserve the model's constraints. No optimizer service is called here; the constraint comparison is a fixed grid sweep. The singularity language is corrected below: the bounded object is the transformed diagnostic, not necessarily the original field.

Pareto Generator Discovery

Each case ranks candidate charts by linear residual and interpretability. The winner is a coordinate system, not a new physical law.

Paper map

Exponential response

y = 2.5 exp(0.8 x)

x chart: identity
y chart: log
slope: 0.800
RMSE: 9.21e-16

The output log chart turns exponential growth into a line with slope 0.8.

Power-law response

y = 3 x^1.7

x chart: log
y chart: log
slope: 1.700
RMSE: 4.63e-16

The log-log chart recovers the scaling exponent as the fitted slope.

Bounded logistic response

y = 1 / (1 + exp(-(1.4 x - 0.3)))

x chart: identity
y chart: logit
slope: 1.400
RMSE: 1.35e-15

The logit output chart reveals the latent linear generator.

Sparse Dynamics Check

A SINDy-style library sees the logistic equation as a two-term polynomial in raw coordinates. The logit coordinate exposes the one-term latent generator.

raw coordinate

x' = 0.9 x - 0.18 x^2

generator coordinate

z' = 0.9, where z = log(x / (5 - x))

The nonlinear logistic ODE becomes a constant-velocity model after the logit coordinate is used.

Reaction Log Coordinates

For positive concentrations, log coordinates turn multiplicative mass-action structure into an additive linear model without breaking positivity.

rate = 0.37 A B

log(rate) = log(0.37) + log(A) + log(B)

A exponent: 1.0
B exponent: 1.0
fit RMSE: 5.4e-16

Constraint-Preserving Latent Maps (grid sweep)

An optimizer can search an unconstrained latent vector while the model receives values already mapped into the physical domain. Below, the same 144-point deterministic grid is evaluated as raw values and through the maps; no optimizer is run.

Positive variable

x = exp(z)

Rates, densities, scales, and stiffness parameters never cross zero.

Bounded probability

p = sigmoid(z)

Probabilities and gates stay in (0, 1) without repair penalties.

Simplex weights

w = softmax(z)

Mixture weights remain nonnegative and sum to one at every evaluation.

grid evaluated as	evaluations	valid	best score	interpretation
raw candidate values	144	13.9%	0.861	Many evaluations are invalid before scoring.
mapped latent values	144	100.0%	0.983	The maps keep candidates in the domain by construction.

Math Framing Corrections

Power-law diagnostic

L_BG[r^n] = n and D_BG[r^n] = exp(n) for r > 0 and positive fields.

Inverse-distance case

L_BG[1/r] = -1, while D_BG[1/r] = exp(-1), not -1.

Singularity language

The transformed diagnostic is bounded; the original field can still diverge.

Direct Sources Read

These external sources anchor the corrected demo framing and the paper expansion.