Discovering How Ice Crystals Grow Using Neural ODE's and Symbolic Regression

TL;DR

This work uses NODE's to learn the functional dependence of unknown physics in the depositional ice growth model by optimizing against experimental measurements of ice crystal mass, and finds a functional form for the depositional ice growth model that best fits 290 mass time series of ice crystals grown in a levitation diffusion chamber.

Abstract

Depositional ice growth is an important process for cirrus cloud evolution, but the physics of ice growth in atmospheric conditions is still poorly understood. One major challenge in constraining depositional ice growth models against observations is that the early growth rates of ice crystals cannot be directly observed, and proposed models require assumptions about the functional dependence of physical processes that are still highly uncertain. Neural ordinary differential equations (NODE's) are a recently developed machine learning method that can be used to learn the derivative of an unknown function. Here we use NODE's to learn the functional dependence of unknown physics in the depositional ice growth model by optimizing against experimental measurements of ice crystal mass. We find a functional form for the depositional ice growth model that best fits 290 mass time series of ice crystals grown in a levitation diffusion chamber. We use symbolic regression to derive an equation for the function learned by the NODE model, which includes additional terms proportional to ice crystal mass in the capacitance growth model. We evaluate this functional form against experimental data sets from the AIDA Aerosol and Cloud Chamber, finding that our new proposed model for depositional ice growth accurately reproduces experimental results in the early stages of ice crystal growth.

Figures (13)

• Figure 1: Overview of methodology for learning unknown physics in depositional ice growth models. Depositional ice growth is an important process for ice formation in atmospheric clouds. Ice crystals grow via direct deposition of water molecules from the vapor phase onto the ice surface. a) We replace partially unknown physics in the depositional ice growth model with a neural network, considering both a strong and weak constraint. b) We integrate the ice growth rate partially parameterized by a neural network and optimize to reduce the distance between the model and experimentally measured time series of ice mass ratios. c) We use symbolic regression to determine a functional form for the unknown physics learned by the neural network.
• Figure 2: Functional dependence of $\alpha$ learned from synthetic data sets. a) Saturation and temperature dependence of $\alpha$ parameterization from Nelson and Baker, 1996 nelson1996new. b) Saturation and temperature dependence from the trained NN for the synthetic data sets. c) Predictions from an expression learned by symbolic regression from the trained NN for the synthetic data sets. d - f) Residuals between $\alpha$ from Nelson and Baker, 1996 and model predictions for $\alpha$.
• Figure 3: Unknown physics learned by weakly-constrained NODE model from experiments. a) The transfer coefficient $G$ learned by the weakly-constrained NODE model for the 290 experiments compared with $G_{c}$ (the transfer coefficient for the continuum case, assuming a spherical ice crystal). b) Predictions from the trained neural network for $G$ compared with the predictions from an expression learned by symbolic regression from the trained neural networks (Eq. 8 in Table S2) for the 290 experiments.
• Figure 4: Comparison of learned depositional ice growth model to a cold cirrus cloud experiment in the AIDA Cloud Chamber. a) Temperature and pressure during the adiabatic expansion experiment. b) Observed supersaturation with respect to ice inside the chamber, compared with predicted supersaturation with respect to ice. Gray dashed line indicates supersaturation equal to 1.0. c) Observed ice water content compared to predicted ice water content. d) Observed ice number density inside of the chamber.
• Figure S1: Overview of experimental data sets used in this analysis.Left: Saturation with respect to ice and temperature at which experiments included in this analysis were performed. The dashed line shows the temperature dependence of saturation with respect to liquid water, while the dotted line show saturation with respect to ice. Symbols indicate whether ice crystals were nucleated heterogeneously or homogeneously. Right: Mass ratios of all ice crystals as a function of time.