{
 "artifacts": null,
 "category": "regularization",
 "description": "Train a neural approximation to a scale-dependent effective action, energy functional, or field while penalizing the residual of a known continuous-symmetry Ward identity. Select the regulator, smoothing scale, or architecture hyperparameter at the minimum Ward residual, and require that the residual decreases when model capacity or derivative-expansion order increases.",
 "formulas_latex": [
  "$$\\delta_{\\mathrm{conf}}f(x)=\\epsilon_\\mu\\left(x^2\\partial_\\mu-2x_\\mu x_\\nu\\partial_\\nu-2x_\\mu D_f\\right)f(x),$$",
  "$$\\mathcal W_\\mu\\Gamma_k=\\Delta_{k,\\mu},$$",
  "$$R(\\theta,\\lambda)=\\frac{\\mathbb E_{x,\\mu}\\left[\\left|\\mathcal W_\\mu\\Gamma_\\theta(x;\\lambda)-\\widehat{\\Delta}_{k,\\mu}(x;\\lambda)\\right|^2\\right]}{\\mathbb E_{x,\\mu}\\left[\\left|\\mathcal W_\\mu\\Gamma_\\theta(x;\\lambda)\\right|^2+\\epsilon\\right]},$$",
  "$$\\lambda^*=\\arg\\min_\\lambda R(\\theta^*(\\lambda),\\lambda),\\qquad \\theta^*(\\lambda)=\\arg\\min_\\theta\\left[L_{\\mathrm{data}}(\\theta)+\\alpha R(\\theta,\\lambda)\\right].$$"
 ],
 "id": 2731,
 "implementation": "1. Exact integration point: represent a scalar effective action, energy, log-density, or coarse-grained field functional with an MLP or neural operator $\\Gamma_\\theta(x;\\lambda)$. Let $\\lambda$ be a Gaussian smoothing width, Fourier cutoff, spectral damping coefficient, or injected-noise scale. Sample coordinates or field configurations x from each training batch and use automatic differentiation to obtain the derivatives in the Ward generator. 2. Pseudocode: compute `gamma = net(x, lambda)`; compute `grad = jacobian(gamma, x)`; calculate `x2 = sum(x*x)` and `xdotgrad = sum(x*grad)`; form `Kmu = x2*grad_mu - 2*x_mu*xdotgrad - 2*x_mu*D`; set `ward = Kmu - Delta_hat`; and optimize `loss = data_loss + alpha*mean(ward**2)/(mean(Kmu**2)+eps)`. 3. Computed versus estimated: the generator and derivatives are computed by autodiff. The cutoff-breaking term $\\widehat{\\Delta}$ should be derived analytically when the discretization or regulator is known; otherwise use zero only when exact symmetry is the target, and report this approximation explicitly. Estimate observable sensitivity by finite differences, $S_Q=|Q(\\lambda+\\delta)-Q(\\lambda-\\delta)|/(2\\delta)$. 4. First cheap experiment: fit a small MLP to synthetic two-dimensional functions constructed to obey scaling or conformal constraints. Compare ordinary MSE against MSE plus Ward loss over ten smoothing widths and widths 32, 64, and 128. The predicted signature is a U-shaped residual curve $R(\\lambda)$ with a stable minimum, reduced observable sensitivity near that minimum, and at least a 20% reduction in R when capacity is increased. If R has no reproducible minimum or does not decrease with capacity, reject the mechanism for the task.",
 "math_summary": "For a scalar function f(x) with scaling dimension D_f, the special conformal generator used in the paper is $\\mathcal K_\\mu=x^2\\partial_\\mu-2x_\\mu x_\\nu\\partial_\\nu-2x_\\mu D_f$. An exactly invariant effective action satisfies $\\mathcal W_\\mu\\Gamma=0$; at finite RG cutoff k and finite truncation, the regulator generates a breaking term, $\\mathcal W_\\mu\\Gamma_k=\\Delta_{k,\\mu}$. For a neural approximation $\\Gamma_\\theta$, define a normalized residual $R$ by comparing its computed Ward operator with the known or estimated breaking term. Here $x$ is the input coordinate or field configuration, $\\mu$ indexes the symmetry generator, $\\theta$ are network parameters, $\\lambda$ is a regulator or smoothing parameter, and $\\epsilon$ prevents division by zero. The transferable criterion is to choose $\\lambda$ where R and the sensitivity of a downstream observable are jointly small, then verify that R falls with increasing truncation capacity.",
 "math_tags": [
  "dynamical-systems",
  "renormalization",
  "geometry",
  "optimization"
 ],
 "ml_areas": [
  "training",
  "regularization",
  "optimizer",
  "mlp"
 ],
 "paper": {
  "arxiv_id": "2608.25103",
  "arxiv_url": "https://arxiv.org/abs/2608.25103",
  "summary_what_math_gives_to_ml": "The paper provides a constructive convergence diagnostic for truncated functional renormalization-group calculations: regulator parameters that minimize conformal Ward-identity breaking also show reduced sensitivity of universal critical exponents. The transferable asset is a computable symmetry-breaking residual for selecting approximation hyperparameters, rather than conformal symmetry by itself. This can be adapted to neural approximations of effective actions, fields, or scale-dependent operators by penalizing violations of known transformation identities and testing whether the residual decreases as model capacity or truncation order increases.",
  "title": "Convergence of the conformal Ward identity in the derivative expansion approximation",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "generalization",
  "accuracy"
 ],
 "title": "Ward-Residual Model Selection",
 "url": "https://synthcore.org/idea/2731/ward-residual-model-selection",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}