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Coupled online learning as a way to tackle instabilities and biases in neural network parameterizations

Stephan Rasp

arXiv:1907.01351v4physics.ao-phphysics.comp-ph

TL;DR

Neural-network parameterizations can produce unstable, biased online simulations, while high-resolution simulations are too expensive for operational forecasting. Coupled online learning adds a training stage that pairs high-resolution targets with states created by the neural-network-driven model; Lorenz 96 experiments support feasibility, but realistic frameworks remain to be tested.

  • Problem

    Machine-learning parameterizations aim to improve subgrid representation, but online simulations have been unstable or biased, and high-resolution models are too expensive for operational forecasting.

    Source: §1 · p. 1

  • Method

    Coupled online learning extends training with high-resolution targets for states produced by the neural-network-driven model, using nudging to keep the high-resolution simulation in sync.

    Source: §3 · p. 5

  • Results

    The Lorenz 96 experiments indicate that coupled learning works well, with the learned neural-network curve gradually approaching the offline curve learned from data generated with the correct parameters.

    Source: §4.3 · pp. 8–9

  • Takeaways & Limitations

    The Lorenz 96 demonstration supports the general feasibility of coupled learning, while testing it in a more realistic framework remains the next step.

    Source: §7 · p. 15

    Lorenz 96 is only a sanity check: it lacks the instabilities and major biases that motivate coupled learning and represents only a small fraction of real-GCM challenges.

    Source: §4.4 · p. 10

Abstract

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Over the last couple of years, machine learning parameterizations have emerged as a potential way to improve the representation of sub-grid processes in Earth System Models (ESMs). So far, all studies were based on the same three-step approach: first a training dataset was created from a high-resolution simulation, then a machine learning algorithm was fitted to this dataset, before the trained algorithm was implemented in the ESM. The resulting online simulations were frequently plagued by instabilities and biases. Here, coupled online learning is proposed as a way to combat these issues. Coupled learning can be seen as a second training stage in which the pretrained machine learning parameterization, specifically a neural network, is run in parallel with a high-resolution simulation. The high-resolution simulation is kept in sync with the neural network-driven ESM through constant nudging. This enables the neural network to learn from the tendencies that the high-resolution simulation would produce if it experienced the states the neural network creates. The concept is illustrated using the Lorenz 96 model, where coupled learning is able to recover the "true" parameterizations. Further, detailed algorithms for the implementation of coupled learning in 3D cloud-resolving models and the super parameterization framework are presented. Finally, outstanding challenges and issues not resolved by this approach are discussed.

1 Introduction

Machine-learning parameterizations could improve subgrid-process representation, but online neural-network simulations have shown instability and bias. The paper proposes coupled online learning and illustrates it with the two-level Lorenz 96 model.

1 Introduction

  • Online neural-network parameterizations have been unstable at times and biased against the reference model, making traditional manual tuning impractical for thousands of network parameters.Subgrid processes, especially clouds, are a major source of uncertainty in climate projections and error in weather predictions.Source: §1 · p. 1
  • Coupled online learning is proposed as a potential way to address instability and bias in machine-learning parameterizations.The paper illustrates the principle using the two-level Lorenz 96 model.Source: §1 · p. 1
  • Figure 1 schematically contrasts machine-learning parameterization workflows with and without coupled online learning.Source: Figure 1 · p. 2

2 Review of online machine learning parameterizations

Prior online machine-learning parameterizations generally follow offline dataset creation and training, then implementation in a model. The reviewed studies report both promising results and challenges, including online instability and drift.

2 Review of online machine learning parameterizations

  • The reviewed workflow creates training data from a reference simulation, trains an algorithm, evaluates it offline, then implements it in the climate model for online integration.Studies of moist convection used simplified aquaplanet settings and omitted some input/output variables, including cloud water and ice.Source: §2 · p. 2

2.1 Rasp et al. (2018) – Super-parameterization with a neural network

  • Rasp et al. (2018) trained a neural network to emulate CRM tendencies in super-parameterization, but online stability was sensitive to small setup changes and biases remained.The study achieved a stable model close to the reference, yet small dataset or input/output changes could lead to unpredictable blow-ups.Source: §2.1 · pp. 2–3

2.2 Brenowitz and Bretherton (2018) – Global 3D CRM with a neural network

  • Brenowitz and Bretherton (2018, 2019) reported online instabilities and drift; later fixes enabled stable longer simulations but did not eliminate drift.The proposed fixes cut upper input levels and used a multi-time-step loss to penalize unstable feedback loops; coarse-graining may contribute to drift.Source: §2.2 · p. 3

2.3 O’Gorman and Dwyer (2018) – Traditional parameterization with a random forest

  • O’Gorman and Dwyer (2018) used random forests, whose predictions are limited by training data and therefore avoid unphysical tendencies that could cause blow-ups.The paper notes that random forests are competitive with neural networks for parameterization, but common implementations do not lend themselves to incremental online learning.Source: §2.3 · p. 4

3 Coupled online learning – the general concept

Coupled online learning runs the ML-driven low-resolution model alongside a nudged high-resolution model, which supplies targets for states generated by the ML model. The paper describes how to derive those targets and update the parameterization.

3 Coupled online learning – the general concept

  • Coupled online learning adds a training stage in which the ML-LR and HR models run in parallel, with the HR state continuously nudged toward the LR state.The nudging timescale controls how closely the models track; overly strong nudging may produce unrealistic HR responses or a blow-up.Source: §3 · p. 4
  • The HR model’s spin-up requirement complicates intermittent supervision and motivates continuously running the HR and LR models in parallel.Source: §3 · p. 4
  • Extending training with HR targets for ML-LR-generated states addresses the problem of neural networks extrapolating beyond the reference training manifold.Source: §3 · p. 5
  • For the 3D HR setup, the target is the HR-internal increment minus the LR increment from non-ML processes; the ML parameterization is then optimized every few time steps.The target derivation assumes nudging and HR-internal evolution are independent over the LR time interval, yielding a linear superposition.Source: §3 · pp. 5–6

4 Parameterization experiments using the Lorenz 96 model

The Lorenz 96 experiments use linear and neural-network parameterizations to illustrate coupled online learning, which works across the tested time-step cases and moves the neural-network fit toward the correct-parameter offline fit. The experiments are a proof of concept: L96 does not exhibit the instability and bias problems that motivate the approach.

4.1 The L96 model

  • The two-level L96 model represents resolved slow variables and unresolved fast variables, with the fast variables evolving ten times faster and at one tenth the amplitude.The experiments use K = 36, J = 10, and a fourth-order Runge–Kutta scheme with time step 0.001.Source: §4.1 · p. 6

4.2 Machine learning parameterizations

  • The experiments compare a linear regression that illustrates the learning procedure with a more realistic neural network, using deterministic parameterizations that are local in space and time.The neural network has two hidden layers of 32 nodes; the linear fit to the illustrated data has a = −0.31 and b = −0.20.Source: §4.2 · pp. 6–7, §4.1 · p. 6

4.3 Coupled online learning10

  • The high-resolution simulation is nudged toward the low-resolution state, while its increment is adjusted to remove the nudging contribution before feature–target samples train the parameterization.The algorithm optimizes squared prediction loss with stochastic gradient descent; the experiments use ten low-resolution steps per neural-network update.Source: §4.3 · p. 8
  • Coupled learning works in both tested time-step cases, with only very small linear-regression intercept differences; the neural-network X–B curve approaches the offline fit learned from correct L96 parameters.With a finer high-resolution time step, the intercept differs slightly from the reference, possibly because the linear-superposition assumption during high-resolution integration is imperfect.Source: §4.3 · pp. 8–9
  • Updating every time step can cause large parameter changes, while collecting samples over several steps smooths convergence; excessively weak nudging can make targets meaningless.For complex Earth system models, strong nudging may pull the high-resolution model too far from its attractor, whereas weaker nudging increases the gap between high- and low-resolution states.Source: §4.3 · pp. 8–9

4.4 Purpose and limitations of L96 experiments

  • L96 is a sanity check, not a test of the motivating instabilities: offline parameterizations are stable and show no major biases in this model.The experiments are proof of concept, and the paper does not exhaustively search hyperparameters; settings for real modeling systems are likely different.Source: §4.3 · pp. 9–10, §4.4 · p. 10

5 Algorithms for online learning in the super-parameterization and 3D HR frameworks

The section describes coupled-learning algorithms for 3D high-resolution models and super-parameterized models. Their learning targets and implementation differ with the model setup.

5.1 3D high-resolution models

  • The 3D HR algorithm uses downscaling and upscaling to transfer states and forcing between model grids, and accumulates gradients rather than storing multiple 3D fields.Its gradient vector has one entry per network parameter and can be computed locally before collection.Source: §5.1 · pp. 10–11
  • In 3D high-resolution models, the network learns a subgrid correction for differences between the LR model and HR dynamics and non-ML physics, not just convection.The approach requires downscaling and upscaling; gradients are computed at each time step and collected for parameter updates every M steps.Source: §5.1 · p. 10

5.2 Super-parameterization

  • In super-parameterization, clean scale separation simplifies learning, and the LR model evolves with ML physics while CRM tendencies provide the learning signal.The CRM and LR model already run in parallel; implementation requires neural-network forward and backward passes, an optimizer, and gradient communication.Source: §5.2 · pp. 11, 13

6 Discussion

The discussion considers which variables to predict, how to impose physical constraints, and practical issues in scaling, implementation, and computational efficiency. It identifies limits and trade-offs in each area.

6.1 Which variables have to be forced/predicted by the neural network?

  • Temperature and humidity may suffice in theory, but omitting information such as cloud water can impair network skill, while physical-budget constraints may require more predicted variables.In SP, only LR prognostic variables are forced during CRM integration; variables such as hydrometeors may evolve without forcing.Source: §6.1 · p. 13

6.2 Physical constraints

  • Hard-coded conservation constraints ensure exact conservation but require predicting all budget variables; omitting terms prevents exact target reproduction but can restrict predictions to conserving solutions.For coarse-grained HR training data, residuals do not themselves obey conservation relations, but LR-grid budget equations can constrain the learned correction.Source: §6.2 · pp. 13–14

6.3 Up- and downscaling

  • HR-to-LR averaging can cause aliasing, while direct LR-to-HR forcing can create sharp boundaries; smoother filtering and interpolation are possible alternatives.How problematic sharp forcing boundaries are in practice remains uncertain.Source: §6.3 · p. 14

6.4 Technical challenges

  • Implementation can require parallel HR and LR models, neural-network passes, gradient communication, and substantial model-specific engineering.The Python-Fortran interface is described as a major obstacle, while calling Python from Fortran may be slow.Source: §6.4 · pp. 14–15

6.5 How efficient is the online learning algorithm?

  • Because HR simulations are expensive, coupled learning must learn efficiently from limited runs; L96 is a poor proxy for judging this because its task and sample size differ substantially.The discussion proposes a dry run with an offline-trained network and a separate, non-shuffled dataset to guide hyperparameter choices and estimate iterations.Source: §6.5 · p. 15

7 Conclusions

The conclusion presents coupled learning as a potential way to address instabilities and tuning, while noting unresolved limits and questions. Its effect on long-term bias remains uncertain because the method can address only short-term prediction errors.

7 Conclusions

  • Coupled learning is presented as a potential way to address instabilities and tuning, but whether short-term error correction reduces long-term biases remains an open question.The conclusion also asks whether the models converge, whether linear superposition holds for large forcing, and how crashes should be handled.Source: §7 · p. 15
  • Generalization remains beyond coupled learning's remit: neural networks should not be expected to perform well outside their training regime.The paper notes that coupled learning can still tune existing parameterizations when they are coded differentiably.Source: §7 · p. 15
  • Deterministic ML models minimizing mean error cannot represent random fluctuations in training data, and how to combine coupled learning with GANs is unclear.The paper also mentions parametric distributions as a possible approach to stochasticity.Source: §7 · pp. 15–16
  • HR simulations are not the truth; observations are the best available account of atmospheric behavior, though they are intermittent and sometimes indirect.Reanalysis is described as a second-best estimate, with exploration of its use for training suggested as worthwhile.Source: §7 · p. 16
  • The paper characterizes coupled online learning as one potential approach among many to overcoming obstacles to improving weather and climate models with ML.It presents the algorithms and challenges and demonstrates general feasibility in L96, with testing in a more realistic framework identified as the next step.Source: §7 · pp. 15–16
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