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# Climate models get memory from simple equations, not neural nets
- URL: https://www.notatechguy.com/climate-models-get-memory-from-simple-equations-not-neural-nets/
- Published: 2026-09-22T07:37:22.000Z
- Updated: 2026-09-22T07:37:23.000Z
- Description: A preprint shows a forced linear ODE recovers most of a neural autoencoder's value for simulating convection in 100-kilometre climate grids.
- Author: Marcello Babbili
- Tags: Technology & AI

Jurij Schönfeld, Tom Beucler and co-authors posted a preprint on arXiv on September 21 showing a simple linear equation can give climate models memory of past weather states [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). The AI methods now used to fill gaps in 100-kilometre-resolution climate grids treat each moment as independent, with no carryover from what came before, which is unrealistic for convection that builds and persists over hours [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com).

**My read:** This is the first climate-modeling paper I've seen that tries to replace a neural network's memory mechanism with a symbolic equation and then checks whether the simpler version still works. The result, that a forced linear ODE recovers "most" but not all of the autoencoder's value, is honest and useful. I'm skeptical of how far this generalises from the Lorenz-96 toy model and offline precipitation tests to a fully coupled climate run. The diurnal cycle result is promising but narrow. What I'd watch is whether any operational weather centre picks this up for an online test.

## Why convection needs a past

Climate models divide the atmosphere into grid boxes roughly 100 kilometres wide [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). Processes smaller than the grid, like individual thunderstorms, happen invisibly inside each box. The model has to guess their collective effect from the box's average conditions. This task is called parameterisation. Recent work trains neural networks to learn these subgrid effects from high-fidelity simulations [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com).

Most of these neural parameterisations are diagnostic. They look at the current state of the grid box and predict what the subgrid processes are doing right now. They have no memory. Yesterday's storms do not influence today's prediction.

Convection has intrinsic persistence, though. A storm that built up over hours does not vanish when the model ticks forward one timestep [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). AI is increasingly being applied to physical systems where getting the dynamics right matters, as when [reinforcement learning cut AI training power violations by 89%](https://www.notatechguy.com/reinforcement-learning-cuts-ai-training-power-violations-89/). Climate parameterisation is the same demand at a different scale.

## From autoencoder to symbolic equation

Schönfeld, Beucler and colleagues, who include Julien Savre at the German Aerospace Center and Steven Sherwood, tackled this in two steps [P²](https://arxiv.org/abs/2609.24882?ref=notatechguy.com). First, they compressed past weather states into a low-dimensional representation using an autoencoder, a neural network that squeezes information into a compact form and then reconstructs it [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). That compressed representation feeds a second neural network that parameterises the subgrid process [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com).

Then comes the twist. They replaced the autoencoder with symbolic equations: a forced multivariate linear ordinary differential equation that governs how the latent variables evolve over time [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). The result is a set of prognostic memory variables, quantities the model can integrate forward alongside the resolved atmospheric state, just like temperature or humidity [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com).

The payoff is interpretability. A linear ODE is something a scientist can read, modify and debug. A neural autoencoder is a black box. The trade-off is performance: the symbolic version recovers most, but not all, of the autoencoder's added value in both test systems [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). The paper does not report exact gap sizes or error bars on that comparison.

Replacing a neural network with a linear equation is the same instinct behind other efficiency gains in ML, as we found when [federated learning privacy error scaling was cut from 4^b to 2^b](https://www.notatechguy.com/federated-learning-privacy-error-scaling-cut-from-4-b-to-2-b/). Simpler math, same job, fewer moving parts.

## Two tests, two settings

The team evaluated the approach on two systems [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). The first was the Lorenz-96 model, a simplified atmosphere analogue, tested online, meaning the parameterisation ran inside the model as it stepped forward in time [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). The second was surface precipitation from high-resolution atmospheric simulations, tested offline, meaning the parameterisation was checked against reference data but not coupled back into a running model [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com).

Against diagnostic parameterisations with no memory, the memory-informed version improved climate statistics and temporal structure [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). One concrete result: it produced a realistic diurnal cycle of tropical land precipitation, the daily rhythm of afternoon storms that diagnostic methods struggle to capture [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com).

## What is and isn't proven

The preprint has not been peer-reviewed [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com). The Lorenz-96 model is a toy system, useful for testing ideas but not a direct proxy for the real atmosphere. The precipitation evaluation was offline, so it does not show the method working inside a fully coupled climate model where errors can compound over decades. Nobody outside the authors' group has replicated the results. The paper states the symbolic approach recovers "most" of the autoencoder's value but does not claim it matches or exceeds the neural version on all metrics [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com).

For a climate modelling group considering this approach, the practical first step would be to take the forced linear ODE form, plug it into an existing convective parameterisation scheme, and run a short online simulation to see whether the diurnal cycle improvement holds when the parameterisation feeds back into the model dynamics. The authors have not published code alongside the preprint, so the equations would need to be reimplemented from the paper.

The next checkpoint for this work is peer review and, ideally, an online test in a full Earth system model, something the authors flag as future work but do not date [S¹](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com).

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*Sources: [S1 — Learning Prognostic Variables for AI Convective Parameterizations via ](https://arxiv.org/abs/2609.24882v1?ref=notatechguy.com) · [P2 — Learning Prognostic Variables for AI Convective Parameterizations via ](https://arxiv.org/abs/2609.24882?ref=notatechguy.com) · [P3 — vanderschaarlab/autoprognosis](https://github.com/vanderschaarlab/autoprognosis?ref=notatechguy.com) · [P4 — Distilling Machine Learning’s Added Value: Pareto Fronts in Atmospheri](https://arxiv.org/html/2408.02161v1?ref=notatechguy.com) · [P5 — peterwestai2/symbolic-knowledge-distillation](https://github.com/peterwestai2/symbolic-knowledge-distillation?ref=notatechguy.com)*

## Related reading

- [Reinforcement learning cuts AI training power violations 89%](https://www.notatechguy.com/reinforcement-learning-cuts-ai-training-power-violations-89/) — our technology desk, 2026-08-13
- [Federated learning privacy error scaling cut from 4^b to 2^b](https://www.notatechguy.com/federated-learning-privacy-error-scaling-cut-from-4-b-to-2-b/) — our technology desk, 2026-08-16
- [OpenAI Academy adds role-based learning paths for five audiences](https://www.notatechguy.com/openai-academy-adds-role-based-learning-paths-for-five-audiences/) — our technology desk, 2026-09-21

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*Written from 5 sourced items, 4 of them primary.*