Solving FRG flow equations without a grid: one network learns the effective potential's temperature, field and scale dependence

Solving functional renormalization group flow equations now has a representation that does not need a grid: one neural network supplies the effective potential's temperature, field and scale dependence together. Earlier physics-informed neural network work took the analytic large-field-number solution as a baseline, trained the network only on finite-field-number corrections, and obtained derivatives by finite differences. In fixed-temperature training at zero temperature, the network's curves agree almost everywhere with the grid method over the renormalization group scale range, with the flow of the quartic coupling as the exception.

The approach writes the solution of the flow equation as an analytic ansatz with tunable parameters and lets the network predict the coefficients inside it rather than the potential itself. The ansatz joins a large-field expansion with a small-field power series, and the network outputs the polynomial coefficients plus a transition scale between the small-field and large-field regions. Training runs on randomly drawn temperature, field and scale triplets, with a loss given by the sum of flow-equation residues at those points and optimized by ADAM followed by L-BFGS.

Enlarging the hidden size raises the accuracy with which the network learns the temperature dependence. At small hidden size the network is not enough to represent the temperature dependence on top of the scale and field dependencies. At larger hidden size, the extrapolated deep-infrared temperature dependence of the order parameter is practically indistinguishable from the grid-based result.

The report covers solving the flow equation of a finite-temperature symmetric scalar field theory in the local potential approximation, with grid and finite-volume results as the comparison. The authors propose that the method extends to systems whose effective potential depends on multiple field variables, and that it could be used to search for renormalization group fixed points in the functional sense, with candidate settings including scalar theories of the relevant symmetry and gauge theories.

The authors report that with varying temperature the same network architecture reaches a terminal loss more than an order of magnitude higher than in fixed-temperature training, so precision is partly sacrificed, and that at small hidden size the network is not enough to represent the temperature dependence. They also note that completely eliminating the unstable region costs accuracy at higher renormalization group scales, which is why an infrared cutoff is introduced, and that maintaining accuracy while lowering that cutoff requires more model parameters, a detailed study left beyond the scope of the work.