a. Numerical advection

1) Numerical scheme and dataset generation

We implemented the so-called L94 advection scheme (Lin et al. 1994) using the Julia scientific computing language (Bezanson et al. 2012). The L94 advection scheme is a second-order accurate van Leer-type advection scheme and has been used in a well-established chemical transport model, GEOS-Chem (Bey et al. 2001), when coupled with a nondirectional multidimensional splitting (Lin and Rood 1996). An implementation in Julia—a differentiable programming language (Innes et al. 2019)—allows us to train machine-learned models by backpropagating error gradients through multiple model time steps. Following Lin et al. (1994), the equations for our 1D numerical advection reference model are as follows.

Suppose we have a scalar field discretized across a 1D grid, with concentration ${\mathrm{\Phi }}_i^n$ in the ith grid point at the nth time step. Then, the cell average value (${\mathrm{\Phi }}_{i+1/2}^n$) can be computed as Eq. (1) because the L94 scheme assumes the piecewise linear distribution of the scalar field inside a cell:

${\mathrm{\Phi }}_{i+1/2}^n=({\mathrm{\Phi }}_i^n+{\mathrm{\Phi }}_{i+1}^n)/2.$(1)

The ultimate purpose of this flux form operator is to calculate the next time cell average as in Eq. (2):

${\mathrm{\Phi }}_{i+1/2}^{n+1}={\mathrm{\Phi }}_{i+1/2}^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}[f_{(i+1)}-f_{(i)}],$(2)

where the flux f_(i) through the cell boundary can be expressed by Eq. (3a) for U_i ≥ 0 or Eq. (3b) for U_i < 0, where U_i is the velocity at the ith edge of the grid cell:

$f_{(i)}=U_i\left[{\mathrm{\Phi }}_{i-1/2}^n+\frac{\mathrm{\Delta }{\mathrm{\Phi }}_{i-1/2}^n}{2}(1-C_i^{-})\right],$(3a)

$f_{(i)}=U_i\left[{\mathrm{\Phi }}_{i+1/2}^n+\frac{\mathrm{\Delta }{\mathrm{\Phi }}_{i+1/2}^n}{2}(1+C_i^{+})\right].$(3b)

Both $C_i^{-}$ and $C_i^{+}$ are Courant–Friedrichs–Lewy (CFL) numbers (Courant et al. 1928), which can be calculated using Eq. (4a) for U_i < 0 and Eq. (4b) for U_i ≥ 0, respectively. Here, Δx_i−1/2 and Δx_i+1/2 are the grid spacing on the left and right sides of the ith gridcell edge and Δt is the time step.

$C_i^{-}=\frac{U_i\mathrm{\Delta }t}{\mathrm{\Delta }{x}_{i-1/2}},$(4a)

$C_i^{+}=\frac{U_i\mathrm{\Delta }t}{\mathrm{\Delta }{x}_{i+1/2}}.$(4b)

The key feature of this reference scheme is the method for deriving the derivative of the cell-averaged scalar, which is implemented using a monotonicity constraint [denoted “mono5” following Lin et al. (1994)] to ensure numerical stability as shown in Eq. (5):

${[\mathrm{\Delta }{\mathrm{\Phi }}_{i+1/2}]}_{\text{mono5}}=\text{sign}({[\mathrm{\Delta }{\mathrm{\Phi }}_{i+1/2}]}_{\text{avg}})\times \text{min}\left[\left|{[\mathrm{\Delta }{\mathrm{\Phi }}_{i+1/2}]}_{\text{avg}}\right|,2({\mathrm{\Phi }}_{i+1/2}-{\mathrm{\Phi }}_{i+1/2}^{\text{min}}),2({\mathrm{\Phi }}_{i+1/2}^{\text{max}}-{\mathrm{\Phi }}_{i+1/2})\right].$(5)

In Eq. (5), ${\mathrm{\Phi }}_{i+1/2}^{\text{max}}$ and ${\mathrm{\Phi }}_{i+1/2}^{\text{min}}$ represent the local maximum and minimum, respectively, of Φ in the three-point stencil centered at i + 1/2. Then, [ΔΦ_i+1/2]_avg is the average of the local spatial difference in Φ as in Eq. (6):

${[\mathrm{\Delta }{\mathrm{\Phi }}_{i+1/2}]}_{\text{avg}}=\frac{(\delta {\mathrm{\Phi }}_i+\delta {\mathrm{\Phi }}_{i+1})}{2},$(6)

where

$\delta {\mathrm{\Phi }}_i={\mathrm{\Phi }}_{i+1/2}^n-{\mathrm{\Phi }}_{i-1/2}^n.$(7)

Our 1D simulation domain is a straight line across the GEOS-FP 0.25° × 0.3125° North America nested grid (NASA 2022) which covers 130°–60°W and 9.75°–60°N. We used the component of the ground-level wind field aligned with the domain line direction—for example, if the line lay in the east–west direction, we would use the east–west component of the given wind field. We use velocity data retrieved from 0.25° × 0.3125° GEOS-FP wind field (NASA 2022) that has been preprocessed for use with GEOS-Chem Classic and is therefore on a rectangular rather than cubed-sphere grid.

We generated a training dataset by simulating advection through the east–west horizontal line on 39.00°N with 0.3125° spatial resolution, using a 300-s time step with the reference solver described above. The thick blue line in Fig. 2 shows the training domain. The simulation period is the first 10 days of January 2019. We fed a square-shaped initial condition which has a 100-ppb mixing ratio on the central one-third of the domain and zero on the rest of the domain. We chose 100 ppb for the initial mixing ratio of the passive tracer as it is a typical mixing ratio for atmospheric pollutants and the default initial mixing ratio of the passive scalar in the GEOS-Chem transport tracer simulation. The square-shaped initial condition results in a challenging training dataset because the integrated wave will have a stiff gradient at the beginning of the simulation, which becomes smoother as the simulation progresses. We specified zero-gradient spatial boundary conditions. This results in simulations with 224 grid cells and 2880 integration time steps.

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Visualization of the 1D training and testing domain.

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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b. Development of the learned advection scheme

1) Model structure

As illustrated in Fig. 3, we designed a physics-informed machine learning solver that is fed with scalar and velocity fields and returns the surrogate coefficients for numerical integration of a three-stencil second-order accurate scheme. We represented the temporal gradient using six terms describing three-stencil first- and second-order derivatives as shown in Eq. (8):

${\mathrm{\Phi }}_i^{n+1}={\mathrm{\Phi }}_i^n+k_1\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}[a_1,a_2,a_3]\cdot [{\mathrm{\Phi }}_{i-1}^n,{\mathrm{\Phi }}_i^n,{\mathrm{\Phi }}_{i+1}^n]+k_2{\left(\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\right)}^2[b_1,b_2,b_3]\cdot [{\mathrm{\Phi }}_{i-1}^n,{\mathrm{\Phi }}_i^n,{\mathrm{\Phi }}_{i+1}^n],$(8)

where k₁ and k₂ are constant gradient-scaling factors (to help with training) and a_i and b_i are the surrogate coefficients generated by the learned solver. This formulation gives our learned scheme the flexibility to use different values of Δx, which can represent the variable relationship between gridcell size in meters and degrees at different latitudes. Overall, Eq. (8) results in a machine-learned model with a strong inductive bias based on the mathematical framework described in section 2a(1).

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Design of the neural-net-based models to emulate the L94 advection solver. The Φⁿ is the scalar field and Uⁿ is the velocity field.

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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We used a 3-layer convolutional neural network (CNN) with 32 filters per layer to fit the surrogate coefficients. Zhuang et al. (2021) used a 4-layer CNN with 32 filters per layer to emulate 1D advection and a 10-layer CNN with 32 filters per layer for 2D. We used 32 filters per layer as they did and reduced the number of layers from 4 to 3 to reduce computational cost. We used the Flux.jl (Innes et al. 2018) machine learning software library to implement the neural network. We trained a separate neural network for each combination of temporal and spatial coarsening. We used a default kernel size of 3, corresponding to a three-stencil scheme, but for simulations where the maximum CFL number is larger than 1, we increased the CNN kernel size (and stencil size) to 5, 9, 17, or 31 to provide a suitable spatiotemporal information horizon. For example, if the maximum CFL number of the subsampled velocity field is between 1 and 2, we set the kernel size to be 5 (i.e., two cells on the left side, one in the center, and two cells on the right side) to allow our algorithm to access information about all scalar values which may pass through the grid cell of interest during a given time step. To facilitate training our neural network across the stiff gradients that characterize this system, we employed the Gaussian error linear unit (GeLU; Hendrycks and Gimpel 2016) activation function as suggested by Kim et al. (2021). For the final activation layer, we used hyperbolic tangent to limit the rate of potential error growth. We chose the hyperbolic tangent function to limit the predicted coefficients to the range −1:1, for consistency with Taylor’s series coefficients that are typically used in advection schemes and also to encourage numerical stability. To normalize the temporal gradient for each term, we multiplied scaling factors on each term [k₁ and k₂ in Eq. (8)], with magnitudes similar to the inverse of Δt/Δx and (Δt/Δx)². In this study, we used 100 for k₁ and 10 000 for k₂, because Δt = 300 s and Δx ≈ 30 000 km in the baseline resolution.

c. Generalization ability testing

We evaluated the generalization ability of the learned schemes against a longer time span (3× longer than the training dataset), different initial conditions (Dirac-delta shape and Gaussian distribution shape), and different wind fields. The tests on the different wind fields can be further categorized by seasons (the first 10 days in April, July, and October to represent spring, summer, and fall, respectively), latitudes (29.5° and 45°N), and domain direction (longitudinal line at 76.875°W using the north–south wind velocity component). The pale blue lines in Fig. 2 show the spatial domains of the generalization tests. For the longer-term stability test, we simulated scalar advection from 1 to 30 January. For other tests, the integration time span is 10 days as in the training data.

Each test has a different purpose. The longer-duration simulations are used to evaluate the degradation of predictive performance over longer time horizons. Changing the shape of the initial conditions tests whether the learned schemes were overfitted to the initial condition shape used in the training dataset. The Dirac delta initial condition test investigates the performance of the learned scheme with extremely large spatial gradients. The purpose of using different velocity fields is to assess the robustness of the learned scheme on the different wind patterns. Finally, the tests with changes in latitude and domain direction evaluate the learned schemes’ ability to work with different grid sizes. For context, the grid size of the training set (Δx, 0.3125° in 39°N) is approximately 27.0 km, while Δx in 29.5° and 45°N is 24.6 and 30.3 km, respectively. The longitudinal grid size, Δy = 0.25°, is approximately 27.8 km at all longitudes.

d. Multidimensional splitting

a. Model performance in emulating the training dataset

Figure 4 summarizes trained model performance in integrating the training dataset with different levels of spatial and temporal coarsening. There are three cases where the learned solver returned unstable outputs, which are discussed in the final paragraph of this section. Except for these unstable cases, the overall errors are fairly small. MAEs and RMSEs range from 2.47–24.8 to 3.94–45.38 ppb, respectively (Figs. 4a,b), and are mostly less than 10 ppb. (For comparison, initial concentrations are 100 ppb.) The R² values are mostly higher than 0.87 with a worst case of 0.70 (Fig. 4c). The most accurate emulation was shown in the 8Δx–16Δt case rather than in the finer grid with a shorter time interval. This is because in the coarser grid, the fine-scale features are averaged out, so the model training is easier than the finer grid. The maximum computational speedup compared to the high-resolution reference model is 18.0× at the coarsest spatial and temporal resolutions. However, under many conditions, our ML model is slower than the reference model owing to the computational intensity of the neural network. We find that temporal coarsening reduces computation time proportionately to the coarsening factor (e.g., 4Δt is 2× faster than 2Δt). Spatial coarsening also reduces computation time but subproportionally to the coarsening factor (e.g., 4Δx is <2× faster than 2Δx) because operations that are sequential in space can leverage the single instruction multiple data (SIMD) compiler optimizations whereas operations that are sequential in time cannot. (The statistics of the computational time by the learned solvers and the reference solvers are given in Tables A2 and A3.)

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Performance of the learned solver in emulating the training dataset at different coarse-graining resolutions as compared to the highest-resolution reference solver simulation. Conditions where integration failed are marked “Unstable.”

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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Computational speedup values reported in Fig. 4 and elsewhere are computed against our Julia implementation of the reference model described in section 2a(1). As a sense check to demonstrate that any ML model speedups that we observed are not owing to an overly slow implementation of the reference model, we compare the execution time of our reference model to that of a GEOS-Chem “transport tracer” configuration, which only computes advection. We turned off other modules including chemistry, depositions, radiation, convection, and boundary layer mixing. On a single CPU in the Illinois Campus Cluster hardware described above, the GEOS-Chem “transport tracer” configuration in the nested 0.25° × 0.3125° North America domain with 4.0° × 5.0° global simulation for providing boundary conditions takes 53 677 s whereas the global 4.0° × 5.0° simulation by itself takes 613 s for a 10-day simulation. Therefore, the computational time for a 10-day simulation in nested North America is to be 53 677 − 613 = 53 065 s or 1.89 μs per grid cell per time step using a single CPU core, whereas our reference solver takes 0.06 μs per grid cell per time step using a single CPU core. Although this comparison does not show that our reference solver is faster than GEOS-Chem—a “transport tracer” simulation does in fact not only do advection so it is not a fair comparison—it does suggest that our reference algorithm is not unreasonably slow.

To evaluate the benefit of our learned advection model, we compare its performance with simulations using the reference method run at spatial and temporal resolutions matching those of the learned models. Figure 5 summarizes the performance of the lower-resolution reference solver. Interestingly, the coarsening with the reference solver appears to have a preferential regime that lies along diagonal elements of the performance matrix (e.g., 2Δx–4Δt and 4Δx–16Δt). Previous research has suggested that shorter time steps unconditionally result in higher model accuracy (Philip et al. 2016), but results here may provide contradictory evidence. Our results instead suggest that matching temporal and spatial coarsening may in some cases enhance the performance of the model as well as the computational speed, possibly because temporal coarsening averages out the high-frequency features which could trigger numerical diffusion in low spatial resolution. Also, the native time step in the GEOS-FP wind field is 60 min, and thus, temporal coarsening from 5 min to a longer step can be beneficial by reducing unnecessary computation and minimizing possible numerical error accumulation. Regardless, additional investigation is required to fully understand the phenomenon we observe here.

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Performance of the reference solver in coarse-graining the fine-resolution training dataset in different resolutions as compared to the highest-resolution reference solver simulation. Conditions where integration failed are marked “Unstable.”

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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When both solvers are run at the same resolution, the computational speed of the reference solver is higher than that of the learned solver, owing to the computational complexity of the neural network (Figs. 4d and 5d). For resolutions where we use a 3-layer CNN, the low-resolution reference solver was 18.7–28.5 times faster than the learned solver. For the (1Δx, 4Δt) and (1Δx, 32Δt) resolutions, we add an additional CNN layer to prevent instability, resulting in learned solvers that are ∼1000× slower than the low-resolution reference solvers. Given the fact that the reference solver with certain coarsening factors such as (8Δx, 64Δt) could achieve much faster speedup than the maximal speedup from the learned solver as well as favorable accuracy, we cannot say that the learned solver in this study is strictly superior to the traditional solver. Instead, we would like to argue that the learned solver presented here could be a complementary option in cases where the traditional solver does not work well. For example, the reference solver does not work well in cases with large grid cells combined with short time steps (Fig. 5)—such as nested simulations with coarse global domains but small time steps to match the higher-resolution inner domains—but the learned solver can still operate with high accuracy in this regime. GEOS-Chem Classic operates the transport with 10 min by default in the global simulation and 5 min by default in the nested simulation. Using our learned solver, we could increase the time interval to even more than 10 min.

The learned solvers with the highest accuracy predicting the training dataset (8Δx, 16Δt) and largest acceleration (16Δx, 64Δt) are shown in Fig. 6, which shows that the spatial patterns with time evolution are well described by the learned schemes in both resolutions. Accuracy is not perfect, and deviations are more noticeable in (16Δx, 64Δt), but R² values combined across all time steps are high in both cases (0.99 and 0.90, respectively).

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Time series representation of the learned emulation of the training dataset. (top row) 8Δx, 16Δt and (bottom row) 16Δx, 64Δt. The initial shapes are slightly different from the square initial condition because of coarse-graining.

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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We are unable to successfully train the model with the coarsening factors of (1Δx, 8Δt), (1Δx, 16Δt), and (1Δx, 64Δt). In these cases, the learned model simulations become numerically unstable. One possible explanation is that those regimes have large CFL numbers due to large temporal coarsening factor relative to that in space. (The low-resolution reference model also becomes unstable under similar conditions.) The maximum CFL numbers in (1Δx, 8Δt), (1Δx, 16Δt), and (1Δx, 64Δt) are 1.82, 3.63, 7.26, and 14.52, respectively. (The maximum CFL numbers in all the tested coarsening cases are summarized in Table A4.) We use larger stencils and larger neural networks in such cases to account for the increased spatiotemporal transport horizon; this solved the problem for (1Δx, 4Δt) and (1Δx, 32Δt) but not for the cases above. Resolving this instability within a computationally efficient framework is a topic for future research.

b. Generalization ability

Figure 7 summarizes error statistics for the generalization tests described in section 2c. (Detailed results for each test are shown in Figs. A2–A6.) The numerical errors in generalization testing datasets are similar to the error in the training dataset in most cases, but outliers exist in the generalization tests with errors substantially larger than that found with the training dataset, and numerical instability does occur in some cases. In most of the unstable cases, we could resolve the instability by using model parameters from a training epoch different than the one that is optimal for the training dataset (those cases are shown in bold letters in Figs. A2–A6). The goal of the analysis here is to explore the performance and flexibility of our method when trained on a small dataset, but we would expect to see better generalization if we used a more diverse training dataset.

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Boxplots of error statistics in generalization tests for (a) MAE, (b) RMSE, and (c) R². Each data point represents one combination of spatial and temporal resolutions. Unstable simulations are not shown here but are shown in Figs. A2–A6.

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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In Fig. 7, the “3 times longer” case shows that predictive performance does not significantly degrade over an extended prediction horizon. It also suggests that the learned solver would work well in integrating wind fields with similar characteristics (e.g., season and latitude) to the training dataset.

The Dirac-delta initial condition test shows that changing the initial condition deteriorates the performance when the new initial condition is extraordinarily stiff (Fig. 7). The normalized error is small, but this is because the initial mass is small, rather than indicating good model performance. The R² value better captures the large uncertainty of the learned emulation in this case. In contrast, the learned solver is robust to the initial condition with a smooth gradient (Gaussian shape; Fig. 7). Zhuang et al. (2021) found their learned 1D advection scheme had a tendency to have a square-shaped scalar even though they fed the Gaussian-distribution-like initial condition because their scheme was trained with the square wave moving to one direction. In our study, that does not happen because our solver could learn from not only the initial shape but also different spatial patterns with time evolution. However, our results still imply that training our model with a wider range of initial conditions would be useful.

The effects of seasonality and different spatial domains are larger than other testing scenarios in terms of median error (Fig. 7). The largest discrepancy in median appears in changing the spatial domain to a longitudinal direction. The largest outlier is seemingly shown in the October wind case, but this is only because Fig. 7 does not show unstable simulations in the July and 29.5°N tests for (2Δx, 64Δt) resolution. As above, we expect that these issues could be resolved by training our model on a more diverse dataset that has different ranges of velocity, stiffness of gradient, etc.

To look closer into the effect of wind velocity range on the generalization skill of learned solvers, we plot MAE versus maximum CFL number of the velocity fields used in the generalization testing (Fig. 8). This plot shows that the learned solver tended to be unstable when the maximum CFL is far from the maximum CFL of the training dataset. The testing cases with the largest discrepancy in maximum CFL with the training dataset show numerical instabilities in (2Δx, 64Δt) resolution. This feature might originate from the structure of our learned solver. As shown in Fig. 3, the learned solver could scale the magnitude of scalar values but not in the case of the velocity field. Therefore, the neural network could fail to give the learned coefficients in a reasonable range when the input velocity is not close to the training regime. Based on this intuition, in future work, we will want to train the solver with a larger dataset that has a wider range of velocity. However, as mentioned above, our goal here is to explore the potential of this approach rather than producing a highly tuned 1D advection solver.

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Dependence of generalization test accuracy on the maximum CFL number of target wind fields. Each boxplot is the summary of errors in each testing scenario integrating wind data different from the training dataset.

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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Overall, the results from our 1D generalization tests suggest that our learned scheme can work in a relatively wide set of circumstances, even when trained on an extremely limited dataset. Our results additionally suggest that it is important for the training dataset to include a distribution of CFL numbers similar to the conditions the model is expected to be deployed under. This will likely be even more crucial in 3D applications where wind speed varies greatly with altitude.

c. 2D demonstration

Figure 9 shows the performance of the learned 1D advection solvers when implemented in a 2D simulation. Even though our learned solvers are trained in 1D, 10 among the 35 2D simulation cases achieve R² values above 0.8. This is a fascinating result given that the velocity fields used in 2D advection have more complex and diverse features than exist in our 1D training dataset. Simply, the 2D advection requires 224 × 201 times of 1D advection, and the relative variety of wind patterns in the 2D dataset could be a function of that dimension. In this regard, our approach looks promising even in 2D applications. The best accuracy is observed in (2Δx, 8Δt), which shows remarkable performance including R² = 1.00. The maximal acceleration is 340 times in the coarsest resolution. (Computational time statistics for 2D advection by the learned solvers are given in Table A5.)

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Performance of the learned solver in 2D implementation in different coarse-graining resolutions. Cases with bold numbers were unstable with the optimal training epoch but stable when using a different training epoch.

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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Although the fact that our 1D solver works at all in a 2D context has exceeded our expectations, we nonetheless proceed with an analysis of strengths and weaknesses with the goal of informing future work. Unlike the results in 1D (Fig. 4), 2D simulations in several resolutions show poor results (R² < 0.5). There are unstable simulations in 2D as well: simulations in (1Δx, 2Δt), (1Δx, 4Δt), (1Δx, 32Δt), and (2Δx, 64Δt) are unstable in 2D advection, though they are stable in 1D. We suspect that this limited performance may be caused by the existence of conditions (e.g., a greater range of wind speeds) that were not present in the training dataset.

To fairly evaluate our approach, we also test the 2D simulations with the low-resolution reference solver. Similar to 1D coarsening cases, 2D advection using the reference solver shows good performance in the diagonal elements of the coarsening performance matrix (Fig. 10). In other coarsening regimes, the reference solvers show a poor R² but usually maintain small errors. This is because the reference solver has the local slope limiter to satisfy the total variance diminishing property, and thus, the simulation is free from spurious oscillation when the simulation does not violate the CFL condition (Durran 2010). Instead, the solution dissipates to zero due to numerical diffusion, resulting in not only low error but also low correlation. Instability occurs when the coarsening results in the CFL number greater than one. Numerical techniques exist that can handle conditions with CFL > 1, but they lie outside of the focus of the current study. The low-resolution reference solvers also achieve very good performance in some resolutions including (2Δx, 4Δt) and (2Δx, 8Δt). The computational speedup by coarsening is always greater with the reference solver than with the learned solver in the same resolution, as explained in section 3a. The maximum acceleration in 2D by the reference solver is 7200×. (Table A6 summarizes the computational time for 2D advection by the reference solvers in different scales.)

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Performance of the reference solver in 2D at different resolutions.

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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We visualize the 2D implementation results of both reference and learned solvers at two resolutions (Fig. 11). We select (2Δx, 8Δt) and (16Δx, 64Δt), representing the best performance and the maximum acceleration by the learned solvers, respectively. In (2Δx, 8Δt), both solvers almost perfectly emulate the finest grid simulation. Both solvers have a tendency to exaggerate the thickness of the plume, but overall they represent the spatial pattern well. For (16Δx, 64Δt), we additionally feed the 1D generalization testing datasets to the model as well as the original training dataset to enhance the learned model. Originally, the learned scheme trained only with the training dataset has limited performance with R² = 0.27 (Fig. 9). With the increased training data, the learned scheme R² is doubled, showing the benefit of introducing more data into the learned model. However, the learned scheme in this resolution overestimates the concentration of the plume and sometimes shows a negative concentration (with a magnitude less than 1 ppb). These artifacts result in a relatively large MAE value. The reference solver in this resolution showed a similar range of R², but dissipates the plume too quickly, and the overall concentration range near the plume is much too low. However, even though the reference solver is dissipative at this resolution, the MAE value was still smaller than the learned solver.

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2D demonstration of coarse-graining by the learned and reference advection schemes. The first row shows the results of the high-resolution baseline solver. The second and fourth rows are the learned advection in different resolutions. The third and fifth rows are the low-resolution reference solvers at different resolutions.

Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0080.1

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As an additional test, we analyzed the power spectra of the simulations shown in Fig. 11 by performing a fast Fourier transform (FFT) in each grid cell and averaging the FFT output over the 2D domain to evaluate the ability of the learned solvers to predict realistic mass transport patterns (shown in Fig. A7). The learned schemes with the native resolution and (2Δx, 8Δt) coarsening reproduced well the power spectra of the reference scheme, while the learned scheme with (16Δx, 64Δt) coarsening correctly represents the shape of the power spectrum distribution while overestimating its amplitude, resulting in the excess dispersion shown in Fig. 11.