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Data driven non-equilibrium moist phase exchanges for atmospheric convection within a discontinuous Galerkin model of the compressible Euler equations

This paper presents a thermodynamically consistent neural network trained on high-resolution convection data to model non-equilibrium moist phase exchanges within a discontinuous Galerkin atmospheric model, demonstrating its ability to simulate sub-km resolution three-phase cloud formation compared to traditional equilibrium-based physics.

Original authors: David Lee, Kieran Ricardo, Junwei Lyu

Published 2026-07-16

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: David Lee, Kieran Ricardo, Junwei Lyu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the atmosphere as a giant, churning kitchen where invisible chefs are constantly cooking up weather. In this kitchen, water isn't just water; it's a shape-shifter that can be a gas (vapor), a liquid (rain or cloud droplets), or a solid (ice). These transformations are the secret sauce behind thunderstorms, hurricanes, and even the gentle rain on a Tuesday. But here's the tricky part: predicting exactly when and how fast water changes from one shape to another is incredibly hard. Traditional weather models try to solve this by using complex math equations that act like a slow, meticulous recipe, calculating every tiny step of the cooking process. While accurate, these recipes are so computationally heavy that they can slow down the supercomputers trying to predict the weather for tomorrow. Scientists are always looking for a faster, smarter way to handle these "phase changes" without losing the flavor of the physics.

This is where a new approach steps in, acting like a culinary apprentice who has watched thousands of cooking videos and learned to guess the next step instantly. A recent study by David Lee and his team at the Bureau of Meteorology and the Australian National University explores using a type of artificial intelligence called a neural network to learn these water transformations. Instead of solving the heavy math equations every time, the AI looks at the current state of the air (how hot, how dense, and how much water is in it) and instantly predicts how much water should turn into rain or ice. The researchers trained this AI on data from a high-resolution weather model that simulates a tropical storm, teaching it the rules of thermodynamics—the laws of heat and energy—so that it doesn't just guess, but guesses in a way that respects the universe's energy balance. The result is a system that simulates the formation of clouds and storms much faster than traditional methods, while still keeping the energy of the system perfectly balanced, ensuring the simulation doesn't "break" the laws of physics.

The Recipe for a Storm

To understand what the researchers did, we first need to look at the problem they are trying to solve. In the real world, when a warm bubble of air rises, it cools down. As it cools, the water vapor inside it wants to turn into liquid water or ice. This process releases heat, which makes the air bubble rise even faster, creating a powerful updraft. This is the engine of a thunderstorm.

In traditional weather models, figuring out exactly how much vapor turns into liquid and how much turns into ice is a headache. The models have to solve a complex puzzle at every single moment in time to ensure that the temperature and pressure are the same for all the water phases (a state called "thermodynamic equilibrium"). It's like trying to balance a scale with a thousand tiny weights on it, over and over again, for every pixel of the weather map. This is accurate, but it's also incredibly slow and expensive for computers to run, especially when trying to predict storms at a very fine, "sub-kilometer" scale where every tiny cloud matters.

The authors of this paper asked a different question: What if we could teach a computer to recognize the pattern of these changes without solving the heavy math puzzle every time? They decided to train a neural network—a digital brain—to learn the "mass exchanges" between vapor, liquid, and ice. Think of the neural network as a student who has watched a master chef (the LFRic model, a sophisticated weather simulator) cook for six hours. The student didn't just memorize the recipe; they learned the feel of the cooking. They learned that when the air gets cold and dense, a certain amount of vapor should turn into liquid, and a bit more might turn into ice, all while keeping the total energy of the kitchen constant.

The Training and the Test

The researchers trained their AI using data from a simulation of a tropical storm over Darwin, Australia. They fed the neural network data on the temperature, density, and amount of water in different forms, and asked it to predict the rate at which water changes from one form to another. To make sure the AI didn't just make things up, they built a special "loss function" into the training. This is like a strict teacher grading the student's homework. The teacher checks two things: first, does the math add up for the mass of water (did we lose any water magically?); and second, does the energy balance out?

The key innovation here is that the AI is designed to be "thermodynamically consistent." This means that even though the AI is making a fast guess, it is forced to follow the rule that energy cannot be created or destroyed. If the AI predicts that vapor turns into ice, it must also calculate exactly how much heat is released and how much the entropy (a measure of disorder or "messiness" in the system) changes to keep the energy equation balanced. The researchers found that they could train the network to an incredibly high level of precision, with the error in its predictions being almost zero (around $6.5 \times 10^{-13}$ ). This tiny number means the AI learned the physics almost perfectly.

The Bubble Experiment

To see if this new AI-driven method actually works in a real weather simulation, the team set up a test case. They created a "moist bubble"—a simulated pocket of warm, moist air rising through a cold atmosphere. They ran this simulation twice: once using their new AI method (which assumes the water phases are not instantly in perfect balance, a state called "chemical dis-equilibrium") and once using the traditional, slow method that forces everything into perfect balance.

The results were fascinating. The bubble simulated with the AI rose and evolved in a way that looked very similar to previous, well-known test cases. It showed a steady conversion of vapor into liquid on the edges of the bubble and into ice in the center. However, the bubble simulated with the traditional "equilibrium" method rose much faster. Why? Because the traditional method assumes that as soon as water freezes, it instantly becomes ice, releasing a huge burst of heat all at once. The AI method, which respects the "non-equilibrium" nature of the real world, showed a more gradual, realistic transition. The water didn't snap into ice immediately; it took its time, releasing heat more slowly.

Crucially, the researchers checked the energy balance. In the AI simulation, the total energy of the system remained perfectly conserved, with errors so small they were essentially non-existent (on the order of $10^{-15}$ ). This proves that the AI isn't just a fast guess; it's a fast, accurate, and physically honest guess. In contrast, the traditional equilibrium method, while fast in its own way, violated the second law of thermodynamics in this specific test, showing a decrease in entropy (a decrease in disorder) when it should have increased, which is physically impossible in an isolated system.

What This Means

The study suggests that using a neural network to handle the complex chemistry of clouds could be a game-changer for weather prediction. By learning the patterns of phase changes from high-resolution data, the AI can simulate storms at a very fine scale without the heavy computational cost of solving complex equilibrium equations at every step. The researchers found that this method respects the laws of physics, keeps energy balanced, and produces results that are more realistic than the "instant balance" assumption used in many current models.

While this is a simulation and not a full-scale weather forecast yet, the results are promising. The authors suggest that in the future, this neural network could replace the heavy microphysics calculations in real-world weather models. This would allow computers to run faster, potentially giving us more accurate and detailed predictions of severe weather. The study also highlights a path forward: ensuring that these AI models always respect the second law of thermodynamics, even when trained on data that might include other factors like boundary conditions. It's a step toward a future where our weather models are not just faster, but smarter and more in tune with the chaotic, beautiful reality of the atmosphere.

Technical Summary: Data-Driven Non-Equilibrium Moist Phase Exchanges

Problem Statement
Convection-resolving atmospheric models require sub-grid parameterizations for microphysical processes involving phase exchanges between vapor, liquid, and ice. Sophisticated multi-moment parameterizations (e.g., CASIM) are computationally expensive because they necessitate the advection of mass and number fractions for each moisture species. Furthermore, traditional implementations often assume mechanical and thermal equilibrium (single temperature and pressure for all phases) while allowing for chemical disequilibrium. The authors seek to address the computational cost of these schemes and the complexity of solving coupled thermodynamic systems by learning the mass exchanges directly from data, while ensuring the resulting model remains thermodynamically consistent and energy-conserving.

Methodology
The study employs a hybrid approach combining data-driven machine learning with a thermodynamically consistent numerical framework:

Data Generation and Training:
- A neural network is trained on output from a regional configuration of the LFRic model using the CASIM multi-moment microphysics parameterization. The dataset covers a 6-hour period of strong tropical convection over Darwin, Australia, at 1.5 km resolution.
- The network architecture is a feed-forward network ( $R^5 \to R^{10} \to R^{60} \to R^{60} \to R^{60} \to R^{12} \to R^3$ ) using leaky ReLU activations.
- Inputs: Thermodynamic state variables derived from prognostic variables: density ( $\rho$ ), entropy ( $\eta$ ), and mass fractions of vapor ( $q_v$ ), liquid ( $q_l$ ), and ice ( $q_i$ ).
- Outputs: Weighted phase exchange rates ( $\tilde{B}, \tilde{C}, \tilde{D}$ ) representing vapor-to-liquid, vapor-to-ice, and liquid-to-ice exchanges.
- Loss Function: The network minimizes the residual of the transport equations for mass fractions. To handle the large disparity in magnitudes between chemical potentials and mass fractions, the transfer terms are scaled. The loss function enforces the conservation of mass fractions as dictated by the governing equations.
Thermodynamically Consistent Formulation:
- The neural network is embedded within a 2D vertical slice discontinuous Galerkin (DG) model of the moist, non-hydrostatic compressible Euler equations.
- The system assumes mechanical and thermal equilibrium (single $T$ and $p$ ) but chemical disequilibrium (different Gibbs free energies).
- Energy Conservation: The formulation utilizes a Hamiltonian structure where the time evolution is driven by skew-symmetric (dynamical) and symmetric (thermodynamic) operators. The change in entropy ( $A$ ) due to phase exchanges is not learned directly but derived analytically from the network's mass exchange outputs to ensure exact energy conservation:
  $A = -\frac{1}{T^2} \left( (\mu_v - \mu_l)^2 B + (\mu_v - \mu_i)^2 C + (\mu_l - \mu_i)^2 D \right)$
- The DG spatial discretization uses third-order polynomials and Gauss-Lobatto-Legendre quadrature. The advection terms are formulated to provably dissipate or conserve tracer variance, enhancing stability.

Key Contributions

Data-Driven Microphysics: The paper demonstrates a method to learn non-equilibrium phase exchanges between vapor, liquid, and ice directly from high-resolution model output, bypassing the need to explicitly advect particle number moments.
Thermodynamic Consistency: A novel integration of a neural network into a compressible Euler solver where the network predicts mass exchanges, and the associated entropy change is derived a posteriori to strictly conserve energy. This avoids the need for iterative Newton problems to match Gibbs free energies at every time step.
Non-Equilibrium Dynamics: The approach explicitly models chemical disequilibrium, allowing for distinct phase behaviors without the instantaneous equilibration assumptions common in traditional schemes.

Results
The authors tested the model using a standard moist bubble test case and compared the non-equilibrium neural network results against a physics-based equilibrium solver:

Bubble Evolution: The non-equilibrium bubble ascended more slowly than the equilibrium bubble. The equilibrium configuration exhibited faster latent heat release and a sharper gradient between liquid and ice phases due to a logical condition forcing all sub-freezing water to be ice.
Phase Conversion: In the non-equilibrium case, approximately one-quarter of the bubble's central mass converted to ice, with steady vapor-to-liquid conversion at the edges. The equilibrium case showed greater total conversion of vapor to liquid and ice.
Thermodynamic Laws: The non-equilibrium simulation respected the second law of thermodynamics, showing positive entropy changes in the absence of diabatic forcing. Conversely, the equilibrium simulation showed negative entropy changes for most of the duration, violating the second law.
Energy Conservation: The sum of power terms associated with phase exchanges and entropy was on the order of $10^{-8}$ W, which is $10^{-15}$ times smaller than the individual exchange powers, indicating energy balance to machine precision. The global energy conservation error converged with the third-order accuracy of the time integrator.

Significance and Claims
The paper claims that this data-driven approach offers a computationally efficient alternative to existing multi-moment parameterizations by negating the need to advect additional moments (such as particle number). Furthermore, the thermodynamically consistent formulation ensures that energy is exactly conserved during phase exchanges, a feature that is difficult to guarantee in standard equilibrium solvers. The authors note that while the network was not explicitly trained to satisfy the second law of thermodynamics, the resulting non-equilibrium configuration naturally adhered to it, unlike the equilibrium benchmark.

The authors suggest that in future work, this neural network could substitute original microphysics parameterizations in regional Numerical Weather Prediction (NWP) models. They posit this could improve efficiency and stability, particularly by ensuring energy conservation in non-equilibrium processes. However, they acknowledge the challenge of ensuring the second law is always respected in the presence of thermal source terms and lateral boundary forcings present in training data, suggesting future work may require global penalty terms or data filtering.

The Recipe for a Storm

The Training and the Test

The Bubble Experiment

What This Means

Technical Summary: Data-Driven Non-Equilibrium Moist Phase Exchanges

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