# Symbolic Regression

In high-barrier cis-trans isomerization the committor, the probability that a configuration reaches product before reactant, is nearly a step function with intermediate values only in a narrow band around the transition state. Combining AIMMD path sampling with symbolic regression, researchers learn the committor from unbiased molecular dynamics trajectories for N-retinylidene-lysine and the p-coumaric acid anion in vacuo and distill it into closed-form expressions. Four proper dihedrals around the reactive bond must couple nonlinearly to reproduce the S-shaped stepwise path seen in the transition-path ensemble, whereas the minimum free-energy path is a straight line.

In high-barrier cis-trans isomerization the committor, the probability that a configuration reaches product before reactant, is nearly a step function with intermediate values only in a narrow band around the transition state. Combining AIMMD path sampling with symbolic regression, researchers learn the committor from unbiased molecular dynamics trajectories for N-retinylidene-lysine and the p-coumaric acid anion in vacuo and distill it into closed-form expressions. Four proper dihedrals around the reactive bond must couple nonlinearly to reproduce the S-shaped stepwise path seen in the transition-path ensemble, whereas the minimum free-energy path is a straight line.

Graph neural networks learn structure–property relationships from graphs of atoms and bonds, but those relationships stay in a latent space and are hard to read as scientific rules. Graph symbolic distillation (GSD) first uses a constrained message-passing architecture search to fix how information is transformed, propagated and aggregated, then fits each layer's learned transformation with a sparse symbolic expression while keeping graph connectivity and aggregation. Across halide-perovskite band gaps, molecular hydration free energies and multidentate adsorption on transition-metal surfaces, it rebuilds expert descriptors from elementary node and edge features and yields testable local-geometry, polarity–size and electronic–connectivity relationships.

Graph neural networks learn structure–property relationships from graphs of atoms and bonds, but those relationships stay in a latent space and are hard to read as scientific rules. Graph symbolic distillation (GSD) first uses a constrained message-passing architecture search to fix how information is transformed, propagated and aggregated, then fits each layer's learned transformation with a sparse symbolic expression while keeping graph connectivity and aggregation. Across halide-perovskite band gaps, molecular hydration free energies and multidentate adsorption on transition-metal surfaces, it rebuilds expert descriptors from elementary node and edge features and yields testable local-geometry, polarity–size and electronic–connectivity relationships.

Computing metallic conductivity from first principles is expensive, so large chemical spaces have gone largely unscreened. This workflow ranks about 2.8 million ordered compounds using two machine-learning proxies, a BoltzTraP2 conductivity under the constant relaxation-time approximation and the Eliashberg electron-phonon coupling λ, combined by symbolic regression into σSR=0.4σBT/λ, then runs full electron-phonon transport on the candidates. Be2IrPt, H3NaNi, LuIr and LiBePt2 reach room-temperature conductivities of roughly 34 to 38×10⁶ S/m, comparable to aluminum's 36.59×10⁶ S/m but still below Ag, Au and Cu.

Computing metallic conductivity from first principles is expensive, so large chemical spaces have gone largely unscreened. This workflow ranks about 2.8 million ordered compounds using two machine-learning proxies, a BoltzTraP2 conductivity under the constant relaxation-time approximation and the Eliashberg electron-phonon coupling λ, combined by symbolic regression into σSR=0.4σBT/λ, then runs full electron-phonon transport on the candidates. Be2IrPt, H3NaNi, LuIr and LiBePt2 reach room-temperature conductivities of roughly 34 to 38×10⁶ S/m, comparable to aluminum's 36.59×10⁶ S/m but still below Ag, Au and Cu.

Effective thermal conductivity of porous silicon has long scattered because pore morphology and substrate doping differ, and no single model covered both. Gas-microphone photoacoustic measurements, atomistic molecular dynamics and physics-informed neural networks were compared on the same 48% to 75% porosity layers: molecular dynamics matched heavily doped columnar structures, while the networks matched the stochastic spongy networks of lightly and moderately doped silicon. Symbolic regression then combined porosity and temperature into one analytical expression, with mean absolute percentage error no more than 6.5% on the training set over 5% to 85% porosity and 300 to 900 K.

Effective thermal conductivity of porous silicon has long scattered because pore morphology and substrate doping differ, and no single model covered both. Gas-microphone photoacoustic measurements, atomistic molecular dynamics and physics-informed neural networks were compared on the same 48% to 75% porosity layers: molecular dynamics matched heavily doped columnar structures, while the networks matched the stochastic spongy networks of lightly and moderately doped silicon. Symbolic regression then combined porosity and temperature into one analytical expression, with mean absolute percentage error no more than 6.5% on the training set over 5% to 85% porosity and 300 to 900 K.

A two-layer, 427k-parameter transformer shows grokking on the finite-field discrete logarithm map: after a long memorization phase it reaches 99.8% held-out accuracy at p = 113. A 12-model capacity sweep from 90k to 14.4M parameters finds grokking recurring at multiple scales, with a non-monotonic region whose outcome changes when training data increase; hidden states grow aligned with multiplicative-character structure, and prime-order and disjoint-generator controls also generalize.

A two-layer, 427k-parameter transformer shows grokking on the finite-field discrete logarithm map: after a long memorization phase it reaches 99.8% held-out accuracy at p = 113. A 12-model capacity sweep from 90k to 14.4M parameters finds grokking recurring at multiple scales, with a non-monotonic region whose outcome changes when training data increase; hidden states grow aligned with multiplicative-character structure, and prime-order and disjoint-generator controls also generalize.

Using neural ordinary differential equations (NODEs) optimized simultaneously across 290 mass-ratio time series of ice crystals grown in a levitation diffusion chamber, the study learned the unknown functional form of the transfer coefficient G in the depositional ice growth model, then used symbolic regression (PySR) to derive a closed-form expression; the weakly-constrained NODE model performed best on 138 of 290 experiments (MSE loss 16495, versus 41396 for no surface kinetics and 40751 for Nelson and Baker 1996), yielding G = a0·Gc^a1·[a2 + a3/m]^-1 + a4 with an added ice-crystal-mass term, and this expression predicted ice supersaturation and ice water content better than the capacitance growth model and Nelson and Baker 1996 in independent AIDA Aerosol and Cloud Chamber experiments.

Using neural ordinary differential equations (NODEs) optimized simultaneously across 290 mass-ratio time series of ice crystals grown in a levitation diffusion chamber, the study learned the unknown functional form of the transfer coefficient G in the depositional ice growth model, then used symbolic regression (PySR) to derive a closed-form expression; the weakly-constrained NODE model performed best on 138 of 290 experiments (MSE loss 16495, versus 41396 for no surface kinetics and 40751 for Nelson and Baker 1996), yielding G = a0·Gc^a1·[a2 + a3/m]^-1 + a4 with an added ice-crystal-mass term, and this expression predicted ice supersaturation and ice water content better than the capacitance growth model and Nelson and Baker 1996 in independent AIDA Aerosol and Cloud Chamber experiments.