# Neural Ordinary Differential Equations

Co2ODE models only the part of a biochemical network that carries signal in a given cellular context: a conditional mask selects the markers tied to rewiring, and a variational autoencoder compresses the rest into a learned closure term trained jointly with a sparse ODE. On single-cell phosphoproteomic data spanning 30 cancer-relevant overexpressions, it compresses a 30-marker panel into a 10-dimensional state while matching the target-trajectory accuracy of a neural ODE fit to the full panel; adding a prior score shifts the selected markers at almost no cost in fit, indicating these data do not identify the true rewiring.

Co2ODE models only the part of a biochemical network that carries signal in a given cellular context: a conditional mask selects the markers tied to rewiring, and a variational autoencoder compresses the rest into a learned closure term trained jointly with a sparse ODE. On single-cell phosphoproteomic data spanning 30 cancer-relevant overexpressions, it compresses a 30-marker panel into a 10-dimensional state while matching the target-trajectory accuracy of a neural ODE fit to the full panel; adding a prior score shifts the selected markers at almost no cost in fit, indicating these data do not identify the true rewiring.

Learning the continuous-time evolution of nonlinear dynamical systems usually relies on trajectory objectives that simulate the learned dynamics during training and backpropagate through the solver, at a cost growing with the rollout horizon. Velocity matching instead regresses the velocity field directly, using piecewise-linear interpolation of observed trajectories and points sampled uniformly within each interval, so no simulation is needed. On nonlinear ODEs and semi-discretized time-dependent PDEs it attains errors more than an order of magnitude lower than trajectory-based objectives under a common iteration budget, and keeps accuracy under schemes and step sizes not used in training.

Learning the continuous-time evolution of nonlinear dynamical systems usually relies on trajectory objectives that simulate the learned dynamics during training and backpropagate through the solver, at a cost growing with the rollout horizon. Velocity matching instead regresses the velocity field directly, using piecewise-linear interpolation of observed trajectories and points sampled uniformly within each interval, so no simulation is needed. On nonlinear ODEs and semi-discretized time-dependent PDEs it attains errors more than an order of magnitude lower than trajectory-based objectives under a common iteration budget, and keeps accuracy under schemes and step sizes not used in training.

DynaVelo puts RNA expression and transcription-factor motif accessibility from single-cell multiome data into one neural differential equation, learning a velocity field that can be perturbed rather than only drawn. In wild-type mouse germinal center B cells it recovers dark-zone cycling, the switch to the light zone and recycling; in Arid1a-heterozygous cells it nominated Ctcf, Stat3 and Bcl6 as rescue targets, and all three double-heterozygous mouse strains restored the dark-zone phenotype.

DynaVelo puts RNA expression and transcription-factor motif accessibility from single-cell multiome data into one neural differential equation, learning a velocity field that can be perturbed rather than only drawn. In wild-type mouse germinal center B cells it recovers dark-zone cycling, the switch to the light zone and recycling; in Arid1a-heterozygous cells it nominated Ctcf, Stat3 and Bcl6 as rescue targets, and all three double-heterozygous mouse strains restored the dark-zone phenotype.

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.