32 modes attains lower frequency errors than the base FNO with 48 modes, highlighting its more efficient representation of complex features while maintaining robustness against overfitting.

<figure><img src="image_19.png" /><figcaption>Figure 15: Comparison of FNO vs. LOGLO-FNO showing 1-step RMSE and bRMSE (↓) on the test set of Kolmogorov Flow ($Re = 5k$) (Li et al., 2022b) for a varying number of modes in the global branch and full set of modes (i.e., (16, 9) for patch size 16 × 16) in the local branch.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Figure 16: Comparison of FNO vs. LOGLO-FNO showing 1-step MELR and WLR (↓) on the test set of Kolmogorov Flow ($Re = 5k$) (Li et al., 2022b) for a varying number of modes in the global branch and full set of modes (i.e., (16, 9) for patch size 16 × 16) in the local branch.</figcaption></figure>

In Figure 16, we plot the energy spectra scalar measures, viz. MELR and WLR, which quantify the spectral discrepancy between the prediction and reference simulations. Here, we observe that LOGLO-FNO is significantly better than baseline FNO.
**Extended Autoregressive Evaluation.** In this inference setup, we evaluate the 1-step trained models on autoregressive rollouts for varying numbers of timesteps and evaluate the rollout errors. We roll out the trajectories until reaching 25 timesteps to understand the behavior of LOGLO-FNO compared to base FNO in terms of autoregressive error accumulation.

In Figures 17, 18, 5, and 19 we plot the autoregressive errors of metrics, such as nRMSE, MaxError, bRMSE, fRMSE, MELR, and WLR. We observe that, overall, LOGLO-FNO consistently yields lower errors compared to base FNO and maintains the same behavior throughout the rollout extent. Although the autoregressive error accumulation is not completely mitigated, we can conclude that LOGLO-FNO suffers less from this problem than the baseline FNO model across all evaluated metrics.

<figure><img src="image_21.png" /><figcaption>Figure 17: Comparison of FNO vs. LOGLO-FNO for autoregressive rollout showing nRMSE and MaxError (↓) growth over varying number of modes (K) in global branch and timesteps in the trajectories on the test set of Kolmogorov Flow ($Re = 5k$) (Li et al., 2022b). The local branch uses a patch size of 16 × 16.</figcaption></figure>

In Figure 19, we visualize the energy spectra deviations through MELR and WLR metrics. We observe that LOGLO-FNO outperforms base FNO by a significant margin, indicating that the deviation or gap between the energy spectra of the ground truth and the predictions of LOGLO-FNO is narrower than that of the baseline FNO model.
<figure><img src="image_22.png" /><figcaption>Figure 18: Comparison of FNO vs. LOGLO-FNO for autoregressive rollout showing RMSE and bRMSE (↓) error growth over varying modes (K) in global branch and number of timesteps in the trajectories on the test set of Kolmogorov Flow ($Re = 5k$) (Li et al., 2022b). The local branch uses a patch size of 16 × 16.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 19: Comparison of FNO vs. LOGLO-FNO for autoregressive rollout showing MELR and WLR (↓) error growth over varying modes (K) in global branch and number of timesteps in the trajectories on the test set of Kolmogorov Flow ($Re = 5k$) (Li et al., 2022b). The local branch uses a patch size of 16 × 16.</figcaption></figure>
G.4.2 Turbulent Radiative Layer 3D

<figure><img src="image_24.png" /><figcaption>Figure 20: Comparison of base FNO vs. LOGLO-FNO showing 1-step spatial domain xRMSE (left) and frequency space fRMSE (right) over varying Fourier modes (K) in the global branch and the local branch utilizing a patch size of 16 × 16 × 32 and the resulting full set of modes (16, 16, 17) on the test set of the Turbulent Radiative Layer 3D dataset (Ohana et al., 2024).</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Figure 21: Comparison of base FNO vs. LOGLO-FNO showing vRMSE values for the autoregressive rollout over the time windows [6:12] (left) and [13:30] (right, both ends inclusive), for variable Fourier modes (K) in the global branch while the local branch employing a patch size of 16 × 16 × 32 and the resulting full set of modes (16, 16, 17) on the test set of the Turbulent Radiative Layer 3D dataset (Ohana et al., 2024).</figcaption></figure>
# H Qualitative Results and Analysis

## H.0.1 Kolmogorov Flow 2D

In this section, we provide qualitative visualizations of the predictions of a few random trajectories from the test set of Kolmogorov Flow 2D dataset (Li et al., 2022b). The predictions are obtained from the models trained with 32 modes in the global branch and a patch size of 16 in the local branch for LOGLO-FNO. Both models use a constant width of 48.

Figure 22 provides a qualitative visualization comparing the ground truth, predictions, and the corresponding absolute errors of two consecutive timesteps of two random trajectories. We observe that LOGLO-FNO retains sharp details, resulting in a noticeable reduction of errors, whereas the base FNO yields smoothed-out predictions. Note that the absolute errors of both predictions are placed on the same scale for a fair comparison.

<figure><img src="image_26.png" /><figcaption>Figure 22: Qualitative comparison of predictions, ground truths, and absolute errors of FNO and LOGLO-FNO for random trajectories from the test set of Kolmogorov Flow ($Re = 5k$) (Li et al., 2022b).</figcaption></figure>

Similar to Figure 22, we visualize the errors of two different trajectories from the test set in Figure 23. However, we additionally localize the regions in the predictions corresponding to regions in the absolute error that exceed 10% of the maximum error in the whole domain. This could result in multiple such regions. Therefore, we limit ourselves to a single contiguous region with the highest intensity in the interest
of providing an uncluttered visualization. The displayed regions inside the red bounding boxes are the ones with the highest error intensity. We observe that the regions overlap for both trajectories at timestep 1. This implies that both models struggle to get the details correct in this region. However, note that the bounding box regions are smaller for LOGLO-FNO compared to base FNO. Hence, we can deduce that LOGLO-FNO is effective in mitigating errors in regions where the base FNO struggles. In scenarios when the bounding boxes do not overlap, as is the case in both model predictions for timestep 2, it is generally desirable to have bounding boxes with smaller areas, as this indicates the errors are localized to a small region and not spread out over a large area.

<figure><img src="image_27.png" /><figcaption>Figure 23: Qualitative comparison of FNO vs. LOGLO-FNO on random samples from the test set of Kolmogorov Flow ($Re = 5k$). The red bounding boxes highlight a single spatially contiguous region where the absolute error exceeds 10% of the maximum error in the domain.</figcaption></figure>

## H.0.2 Turbulent Radiative Layer 3D

In this section, we provide qualitative visualizations of predictions of a few random trajectories from the test set of the Turbulent Radiative Layer 3D dataset (Ohana et al., 2024). The LOGLO-FNO and base FNO predictions are obtained using 1-step evaluation. In the following plots, we show the test set trajectory corresponding to the $t_{cool}$ parameter 0.03. The x, y, and z axes show the spatial extents ($L_x, L_y$, and $L_z$) of the simulation. The ground truth solution is repeated in the left column for ease of comparison and clarity in the representation. Note the deviation in the range of density and velocity values predicted by base FNO and LOGLO-FNO with respect to the ground truth. Further, we observe from Figure 24 that base FNO predicts unphysical values (i.e., negative density), whereas the density values predicted by LOGLO-FNO are non-negative.
<figure><img src="image_28.png" /><figcaption>Figure 24: A cross-sectional (XY) slice at the middle of the Z-axis, embedded in the XY, YZ, and XZ planar view, showing the *density* channel from the test set trajectory (for *t<sub>cool</sub>* = 0.03) at timestep 5 comparing the ground truth (left), LOGLO-FNO and base FNO predictions (right).</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>Figure 25: A cross-sectional (XY) slice at the middle of the Z-axis, embedded in the XY, YZ, and XZ planar view, showing the *velocity-z* channel from the test set trajectory (for *tcool* = 0.03) at timestep 5 comparing the ground truth (left), LOGLO-FNO and base FNO predictions (right).</figcaption></figure>
# I Energy Spectra Visualization and Analysis

**Kolmogorov Flow.** In this section, we provide a comparative analysis of the energy spectra of predictions of state-of-the-art neural operator baselines versus the reference solution $u$ on the Kolmogorov Flow 2D dataset. The predictions are obtained by autoregressive rollout of the 1-step trained models for the entire duration of the trajectory. In Figures 26, 27, 28, 29, 30, 31, 32, and 33, we plot the energy spectra for two consecutive timesteps at 50 timestep intervals.

<figure><img src="image_30.png" /><figcaption>Figure 26: Energy spectra comparison of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth at two consecutive timesteps on a random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK$^{†}$ denotes the use of both local integral and differential kernels for local convolutions.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Figure 27: Energy spectra comparison of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth at two consecutive timesteps on a random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK$^{†}$ indicates the use of both local integral and differential kernels for local convolutions.</figcaption></figure>
<figure><img src="image_32.png" /><figcaption>Figure 28: Energy spectra comparison of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth at two consecutive timesteps on a random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK$^{†}$ denotes the use of both local integral and differential kernels for local convolutions.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>Figure 29: Energy spectra comparison of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth at two consecutive timesteps on a random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK$^{†}$ indicates the use of both local integral and differential kernels for local convolutions.</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>Figure 30: Energy spectra comparison of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth at two consecutive timesteps on a random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK$^{†}$ represents the use of both local integral and differential kernels for local convolutions.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>Figure 31: Energy spectra comparison of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth at two consecutive timesteps on a random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK$^{†}$ indicates the use of both local integral and differential kernels for local convolutions.</figcaption></figure>
<figure><img src="image_36.png" /><figcaption>Figure 32: Energy spectra comparison of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth at two consecutive timesteps on a random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK$^{†}$ denotes the use of both local integral and differential kernels for local convolutions.</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>Figure 33: Energy spectra comparison of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth at two consecutive timesteps on a random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK$^{†}$ indicates the use of both local integral and differential kernels for local convolutions.</figcaption></figure>
# J Analysis of Correlation with Ground Truth on AR Rollout

## J.1 Pearson's Correlation of Sample Trajectories with Ground Truth

In this section, we first describe the steps for Pearson's correlation coefficient computation and then provide visualizations of Pearson's correlation of the baselines and our proposed LOGLO-FNO model predictions with respect to the reference solution on randomly sampled trajectories from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a).

Let $X \in \mathbb{R}^{N_b \times N_t \times N_x \times N_y}$ and $Y \in \mathbb{R}^{N_b \times N_t \times N_x \times N_y}$ represent the predictions from a model and the ground truth, respectively, where:

* $N_b$ is the batch size and $N_t$ is the sequence length or the number of time frames,

* $N_x$ and $N_y$ are the resolutions of spatial dimensions (e.g., width and height for 2D data),

We compute the Pearson correlation coefficient between $X$ and $Y$ in a vectorized manner as follows.

**Step 1: Reshape Tensors.** The tensors $X$ and $Y$ are reshaped by flattening the spatial dimensions $N_x$ and $N_y$ into a single dimension $S = N_x \times N_y$.

$$X = X_{\text{reshape}} \in \mathbb{R}^{N_b \times N_t \times S}, \quad Y = Y_{\text{reshape}} \in \mathbb{R}^{N_b \times N_t \times S}.$$

**Step 2: Compute Mean.** The mean values of $X$ (predictions) and $Y$ (ground truth) along the spatial dimension $S$ are computed as,

$$\mu_X = \frac{1}{S} \sum_{i=1}^{S} X_{:,:,i},$$

$$\mu_Y = \frac{1}{S} \sum_{i=1}^{S} Y_{:,:,i},$$

where:

* $\mu_X \in \mathbb{R}^{N_b \times N_t}$ and $\mu_Y \in \mathbb{R}^{N_b \times N_t}$ are the mean tensors for $X$ and $Y$, respectively,

* The colon notation $:,:$, $i$ indicates that the operation is performed over the spatial dimension $S$ for each sample in the batch $N_b$ and for each time step in $N_t$.

**Step 3: Compute Standard Deviation.** The standard deviations of $X$ (predictions) and $Y$ (ground truth) along the spatial dimension $S$ are computed as,

$$\sigma_X = \sqrt{\frac{1}{S} \sum_{i=1}^{S} (X_{:,:,i} - \mu_X)^2},$$

$$\sigma_Y = \sqrt{\frac{1}{S} \sum_{i=1}^{S} (Y_{:,:,i} - \mu_Y)^2},$$

where $\sigma_X \in \mathbb{R}^{N_b \times N_t}$ and $\sigma_Y \in \mathbb{R}^{N_b \times N_t}$ are the standard deviations for $X$ and $Y$, respectively.

**Step 4: Compute Covariance.** The covariance between $X$ (predictions) and $Y$ (ground truth) can then be computed as,

$$\text{cov}(X, Y) = \frac{1}{S} \sum_{i=1}^{S} (X_{:,:,i} - \mu_X) \odot (Y_{:,:,i} - \mu_Y),$$

where $\text{cov}(X, Y) \in \mathbb{R}^{N_b \times N_t}$ is the covariance tensor and $\odot$ denotes element-wise multiplication.
**Step 5: Compute Pearson's Correlation Coefficient.** The Pearson's correlation coefficient for each sample in the batch $N_b$ and time step $N_t$ is computed using,

$$ \operatorname{corr}(\mathbf{X}, \mathbf{Y}) = \frac{\operatorname{cov}(\mathbf{X}, \mathbf{Y})}{\sigma_{\mathbf{X}} \odot \sigma_{\mathbf{Y}}} $$

To ensure numerical stability, the denominator is clamped to a small positive value $\epsilon$ (for instance, set $\epsilon = \text{torch.finfo(torch.float32).tiny}$):

$$ \operatorname{corr}(\mathbf{X}, \mathbf{Y}) = \frac{\operatorname{cov}(\mathbf{X}, \mathbf{Y})}{\max(\sigma_{\mathbf{X}} \odot \sigma_{\mathbf{Y}}, \epsilon)} $$

where $\operatorname{corr}(\mathbf{X}, \mathbf{Y}) \in \mathbb{R}^{N_b \times N_t}$ is the output tensor containing the Pearson's correlation coefficients for each sample in the batch and time step.

**Interpretation.** Pearson's correlation coefficient measures the linear relationship between the model predictions $\mathbf{X}$ and the ground truth solution $\mathbf{Y}$. Therefore, a value of

* 1 indicates a perfect positive linear relationship,

* -1 indicates a perfect negative linear relationship, and

* 0 indicates no linear relationship.

<figure><img src="image_38.png" /><figcaption>Figure 34: Comparison of Pearson's correlation of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth on a random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK† indicates the use of both local integral and differential kernels for local convolutions.</figcaption></figure>
<figure><img src="image_39.png" /><figcaption>Figure 35: Comparison of Pearson's correlation of predictions of state-of-the-art neural operator baselines and our LOGLO-FNO vs. ground truth on a different random sample from the test set of Kolmogorov Flow 2D ($Re = 5k$) (Li et al., 2022a). NO-LIDK$^{†}$ indicates the use of both local integral and differential kernels for local convolutions.</figcaption></figure>
# K Ablation Studies

Through the following ablation studies, we attempt to ascertain the impact and importance of each of the introduced components (i.e., local spectral convolution branch and high-frequency propagation module) as well as the patch size sensitivity, high-frequency adaptive Gaussian noise, and radially binned frequency-aware loss term.

## K.1 LOGLO-FNO Devoid of Radially Binned Frequency Loss ($C_{freq}$)

We conduct an ablation study to assess the relative importance of the proposed radially binned spectral energy of errors as a frequency-aware loss term compared to the architectural modifications by the introduction of layers for local spectral convolution and high-frequency propagation. Therefore, we train LOGLO-FNO by retaining only the local branch and high-frequency propagation module in parallel to the global branch. By setting $C_{freq} = 0$, the cost function $C$ for model training then reduces to the plain MSE loss.

$$ \theta^* = \arg \min_{\theta} \sum_{n=1}^{N} \sum_{t=1}^{T-1} C_{MSE}(\mathcal{N}_{\theta}(u^t), u^{t+1}) $$

The results of this study on the Turbulent Radiative Layer 3D dataset are provided in Table 7 on row 3.

## K.2 LOGLO-FNO Devoid of HFP Module and Radially Binned Frequency Loss (HFP, $C_{freq}$)

In this ablation setup, we remove the high-frequency propagation module and keep only the local branch. The radially binned frequency loss term is also disabled by setting $C_{freq} = 0$. Therefore, the LOGLO-FNO architecture comprises only the local and global branches in the Fourier layers, and is trained with plain MSE.

The results of this ablation choice on Turbulent Radiative Layer 3D are provided in Table 7 on row 4.

## K.3 LOGLO-FNO Devoid of High Frequency Propagation Module (HFP)

This ablation setup is similar to the previous study in §K.2, except that we now enable the radially binned frequency loss in the training objective to isolate the effect of removing the HFP module. The 1-step evaluation results for this model setup on Turbulent Radiative Layer 3D are placed in the last row of Table 7.

Table 15: 1-step evaluation of LOGLO-FNO on the test set of Turbulent Radiative Mixing Layer 3D dataset (Fielding et al., 2020; Ohana et al., 2024). Base FNO uses a spectral filter size of 18 and 48 hidden channels. LOGLO-FNO uses 16 and (16, 16, 17)\* Fourier modes in the global and local branches, respectively, whereas the width is set to 52. (\*This is because the Turbulent Radiative Layer 3D dataset has 2× more sampling points on the z-axis relative to the x- and y-axes resolutions – (128 × 128 × 256).)

<table><thead><tr><th>Model</th><th>RMSE (↓)</th><th>nRMSE</th><th>bRMSE</th><th>cRMSE</th><th>fRMSE(L)</th><th>fRMSE(M)</th><th>fRMSE(H)</th><th>vRMSE (↓)</th><th>MaxError (↓)</th></tr></thead><tbody><tr><td colspan="10" style="text-align: center; font-weight: bold;">1-step Evaluation</td></tr><tr><td>Base FNO</td><td>2.76 · 10<sup>-1</sup></td><td>2.97 · 10<sup>-1</sup></td><td>6.83 · 10<sup>-1</sup></td><td>2.3 · 10<sup>-2</sup></td><td>5.17 · 10<sup>-2</sup></td><td>4.45 · 10<sup>-2</sup></td><td>2.76 · 10<sup>-2</sup></td><td>3.09 · 10<sup>-1</sup></td><td>1.11 · 10<sup>1</sup></td></tr><tr><td>LOGLO-FNO<br>(+freq loss, +HFP)</td><td>2.42 · 10<sup>-1</sup></td><td>2.58 · 10<sup>-1</sup></td><td>6.19 · 10<sup>-1</sup></td><td>1.75 · 10<sup>-2</sup></td><td>4.1 · 10<sup>-2</sup></td><td>3.62 · 10<sup>-2</sup></td><td>2.45 · 10<sup>-2</sup></td><td>2.68 · 10<sup>-1</sup></td><td>1.06 · 10<sup>1</sup></td></tr><tr><td>LOGLO-FNO<br>(-freq loss, +HFP)</td><td>2.57 · 10<sup>-1</sup></td><td>2.75 · 10<sup>-1</sup></td><td>6.51 · 10<sup>-1</sup></td><td>1.85 · 10<sup>-2</sup></td><td>4.4 · 10<sup>-2</sup></td><td>4.09 · 10<sup>-2</sup></td><td>2.63 · 10<sup>-2</sup></td><td>2.86 · 10<sup>-1</sup></td><td>1.09 · 10<sup>1</sup></td></tr><tr><td>LOGLO-FNO<br>(-freq loss, -HFP)</td><td>2.58 · 10<sup>-1</sup></td><td>2.76 · 10<sup>-1</sup></td><td>6.55 · 10<sup>-1</sup></td><td>1.83 · 10<sup>-2</sup></td><td>4.4 · 10<sup>-2</sup></td><td>4.13 · 10<sup>-2</sup></td><td>2.64 · 10<sup>-2</sup></td><td>2.87 · 10<sup>-1</sup></td><td>1.1 · 10<sup>1</sup></td></tr><tr><td>LOGLO-FNO<br>(+freq loss, -HFP)</td><td>2.47 · 10<sup>-1</sup></td><td>2.66 · 10<sup>-1</sup></td><td>6.32 · 10<sup>-1</sup></td><td>2.03 · 10<sup>-2</sup></td><td>4.53 · 10<sup>-2</sup></td><td>3.76 · 10<sup>-2</sup></td><td>2.48 · 10<sup>-2</sup></td><td>2.76 · 10<sup>-1</sup></td><td>1.07 · 10<sup>1</sup></td></tr></tbody></table>

As we observe from Table 7, our model achieves the lowest errors across all metrics when the local branch is combined with both the radially binned frequency loss and the HFP module.
## K.4 Patch-Size Sensitivity of LOGLO-FNO

In this section, we first provide the results of an ablation study investigating the sensitivity of LOGLO-FNO on the patch sizes for the local branch through an experiment on the Turbulent Radiative Layer 3D PDE problem. Across the tested patch size variations ($p_{size} = \{4, 8, 16\}$), only the patch size values are altered while all other hyperparameters, including the learning rate and random seed, remain the same during training.

<figure><img src="image_40.png" /><figcaption>Figure 36: Effect of patch sizes on the Time-Averaged vRMSE and 1-step spatial and spectral evaluation metrics on the Turbulent Radiative Layer 3D dataset.</figcaption></figure>

In Figure 36, we visualize the Time-Averaged vRMSE and 1-step xRMSE evaluation scores for patch sizes 4, 8, and 16. While we observe a trend of minor improvement in 1-step errors with the increase of patch sizes, the Time-Averaged autoregressive errors do not exhibit this clear trend. We note from Figure 36 that an increase in patch size directly results in an increased number of trainable parameters (and by virtue of it, increases model expressivity) since we use the full set of Fourier modes in the local branch. Therefore, as a trade-off between errors and training time or memory usage, we use a patch size of 16.

As an additional case study on patch size sensitivity analysis, we conduct experiments on the Compressible Euler four-quadrant Riemann problem 2D from Poseidon (Herde et al., 2024), and provide the 1-step and autoregressive error evolution of overall (nRMSE) and frequency-decomposed (fRMSE) evaluations over an increasing number of timesteps for four different patch size configurations of LOGLO-FNO in Figures 37 and 38, respectively.

<figure><img src="image_41.png" /><figcaption>Figure 37: Effect of patch sizes on the 1-step spatial (left: nRMSE and vRMSE) and spectral (right: fRMSE) evaluation metrics on the Compressible Euler four-quadrant Riemann problem 2D (Herde et al., 2024).</figcaption></figure>

Shifting our attention to the rollout evaluations, we note from Figure 38 a trend of error improvements as we increase the patch sizes from 4 to 16. However, a further increase in patch size to 32 does not yield an improvement, but rather detrimental, over long rollout scenarios. Therefore, as was previously found in the
case of the Turbulent Radiative Layer 3D problem, we come to the conclusion that a patch size of 16 is a
sufficiently good choice.

<figure><img src="image_42.png" /><figcaption>Figure 38: Error accumulation and patch size sensitivity during autoregressive rollouts on the Compressible Euler four-quadrant Riemann problem 2D (Herde et al., 2024).</figcaption></figure>
## K.5 Influence of High-Frequency Adaptive Gaussian Noise on Rollout Stability in LOGLO-FNO

In this section, we conduct an ablation study to investigate the influence of high-frequency adaptive Gaussian noise (HF-AGN) in LOGLO-FNO on the stability of trajectory rollouts for the Kolmogorov Flow 2D (Re5000) dataset. We report the mean and standard deviation values of errors at timestep 400 over three runs in Table 16. The results suggest that HF-AGN is helpful for the stability of longer rollout scenarios.

<table><caption>Table 16: Rollout errors of LOGLO-FNO, with and without adaptive Gaussian noise (HF-AGN) injected during 1-step training, at timestep 400 on the Kolmogorov Flow 2D (Re5000) dataset. The autoregressive evaluation is over three runs.</caption><thead><tr><th>Group</th><th>RMSE (↓)</th><th>nRMSE</th><th>bRMSE</th><th>cRMSE</th><th>fRMSE(L)</th><th>fRMSE(M)</th><th>fRMSE(H)</th><th>MaxError (↓)</th><th>MELR (↓)</th><th>WLR (↓)</th></tr></thead><tbody><tr><td>LOGLO-FNO<br>(-HF-AGN)</td><td>7.8891<br>(±1.2168)</td><td>1.4572<br>(±0.2319)</td><td>7.5518<br>(±1.0478)</td><td>0.0472<br>(±0.0614)</td><td>1.4525<br>(±0.2229)</td><td>0.8524<br>(±0.2458)</td><td>0.1726<br>(±0.0227)</td><td>66.2614<br>(±13.5936)</td><td>0.4811<br>(±0.0662)</td><td>0.2797<br>(±0.0699)</td></tr><tr><td>LOGLO-FNO<br>(+HF-AGN)</td><td><b>7.1982</b><br>(±0.1837)</td><td><b>1.3227</b><br>(±0.0337)</td><td><b>7.0794</b><br>(±0.2293)</td><td><b>0.0113</b><br>(±0.0050)</td><td><b>1.2965</b><br>(±0.0512)</td><td><b>0.7362</b><br>(±0.0151)</td><td><b>0.1667</b><br>(±0.0012)</td><td><b>58.9953</b><br>(±1.7618)</td><td><b>0.4061</b><br>(±0.0795)</td><td><b>0.2228</b><br>(±0.0070)</td></tr></tbody></table>

## K.6 Choice of Radial Bin Cutoff Values for the Radially Binned Frequency Loss in LOGLO-FNO

In order to assess the sensitivity of the cutoff boundaries when demarcating the low, mid, and high frequency regions for the radially binned spectral loss, we conduct an ablation study on the 2D Kolmogorov Flow dataset and provide the 5-step AR errors in Table 17.

We note that the choice iLow=2 and iHigh=10 achieves the lowest spatial (nRMSE) and low-frequency errors and ties with the iLow=4 and iHigh=12 setup on the high-frequency error. However, the latter might be an optimal choice since it exhibits the lowest variance on the spatial (nRMSE), mid-, and high-frequency errors and is more stable over long rollouts.

<table><caption>Table 17: Sensitivity analysis of radial bin cutoffs for the radially binned frequency loss applied for training LOGLO-FNO. The experiments are on the Kolmogorov Flow 2D (Re5000) dataset, and the evaluations are over five independent runs with unique random seeds.</caption><thead><tr><th>Cutoff Boundary</th><th>Unpenalized Region</th><th>Penalized Region</th><th>nRMSE (↓)</th><th>fRMSE(L)</th><th>fRMSE(M)</th><th>fRMSE(H) (↓)</th></tr></thead><tbody><tr><td>iLow=2, iHigh=10</td><td>r &lt; 2</td><td>r ≥ 2</td><td><b>0.1917</b><br>(± 0.0039)</td><td><b>0.0262</b><br>(± 0.0008)</td><td>0.0466<br>(± 0.0025)</td><td><b>0.0694</b><br>(± 0.0014)</td></tr><tr><td>iLow=4, iHigh=12</td><td>r &lt; 4</td><td>r ≥ 4</td><td>0.1931<br>(± 0.0021)</td><td>0.0285<br>(± 0.0025)</td><td><b>0.0462</b><br>(± 0.0020)</td><td><b>0.0694</b><br>(± 0.0008)</td></tr><tr><td>iLow=6, iHigh=15</td><td>r &lt; 6</td><td>r ≥ 6</td><td>0.1942<br>(± 0.0034)</td><td>0.0281<br>(± 0.0010)</td><td>0.0485<br>(± 0.0025)</td><td>0.0702<br>(± 0.0014)</td></tr></tbody></table>

<footnote>Of the total four runs we conducted, we exclude a run from the mean computation due to a blow-up in errors for the model without HF-AGN.</footnote>
## L Zero-Shot Super-Resolution (ZSSR) Capability and Performance of LOGLO-FNO

In this section, we evaluate the capability and performance of LOGLO-FNO on spatial ZSSR tasks and compare it with the results of NO-LIDK (Liu-Schiaffini et al., 2024) and the baseline FNO model.

<table><caption>Table 18: ZSSR evaluation of LOGLO-FNO on the test set of Kolmogorov Flow 2D dataset (Li et al., 2022b). Base FNO and NO-LIDK* use a spectral filter size of 12 and 20 hidden channels. NO-LIDK* denotes using only localized integral kernel and NO-LIDK† means employing both disco and diff layers. LOGLO-FNO uses 12 and (16, 9) Fourier modes in the global and local branches, respectively, whereas the width is set to 20. Training spatial resolution is (64 × 64) and denoted as 1×, whereas the ZSSR test resolution is (128 × 128) and denoted as 2×. We boldface the lowest errors and underline the value when the 2^nd^ best model is LOGLO-FNO.</caption><thead><tr><th>Model</th><th>Resolution</th><th>nRMSE (↓)</th><th>fRMSE(L)</th><th>fRMSE(M)</th><th>fRMSE(H)</th><th>MELR (↓)</th><th>WLR (↓)</th></tr></thead><tbody><tr><td colspan="8" style="text-align:center; font-weight:bold;">1-step Evaluation</td></tr><tr><td rowspan="2">Base FNO</td><td>1×</td><td>2.99 · 10<sup>-1</sup></td><td>4.71 · 10<sup>-2</sup></td><td>8.3 · 10<sup>-2</sup></td><td>1.59 · 10<sup>-1</sup></td><td>6.41 · 10<sup>-1</sup></td><td>1.2 · 10<sup>-1</sup></td></tr><tr><td>2×</td><td>2.93 · 10<sup>-1</sup></td><td><b>3.14</b> · 10<sup>-2</sup></td><td>7.38 · 10<sup>-2</sup></td><td>1.16 · 10<sup>-1</sup></td><td>9.61 · 10<sup>-1</sup></td><td>1.43 · 10<sup>-1</sup></td></tr><tr><td rowspan="2">NO-LIDK*</td><td>1×</td><td>2.79 · 10<sup>-1</sup></td><td>4.65 · 10<sup>-2</sup></td><td>8.88 · 10<sup>-2</sup></td><td>1.41 · 10<sup>-1</sup></td><td>5.89 · 10<sup>-1</sup></td><td>1.04 · 10<sup>-1</sup></td></tr><tr><td>2×</td><td>3.71 · 10<sup>-1</sup></td><td>5.16 · 10<sup>-2</sup></td><td>7.87 · 10<sup>-2</sup></td><td>1.12 · 10<sup>-1</sup></td><td>9.21 · 10<sup>-1</sup></td><td>1.45 · 10<sup>-1</sup></td></tr><tr><td rowspan="2">NO-LIDK†</td><td>1×</td><td>2.56 · 10<sup>-1</sup></td><td>4.61 · 10<sup>-2</sup></td><td>7.77 · 10<sup>-2</sup></td><td>1.28 · 10<sup>-1</sup></td><td>2.32 · 10<sup>-1</sup></td><td>5.89 · 10<sup>-2</sup></td></tr><tr><td>2×</td><td>4.35 · 10<sup>-1</sup></td><td>1.11 · 10<sup>-1</sup></td><td>1.75 · 10<sup>-1</sup></td><td>1.39 · 10<sup>-1</sup></td><td>2.6 · 10<sup>-1</sup></td><td>6.6 · 10<sup>-2</sup></td></tr><tr><td rowspan="2">LOGLO-FNO</td><td>1×</td><td><b>2.32</b> · 10<sup>-1</sup></td><td><b>4.57</b> · 10<sup>-2</sup></td><td><b>7.35</b> · 10<sup>-2</sup></td><td><b>1.22</b> · 10<sup>-1</sup></td><td><b>2.12</b> · 10<sup>-1</sup></td><td><b>4.68</b> · 10<sup>-2</sup></td></tr><tr><td>2×</td><td><b>2.9</b> · 10<sup>-1</sup></td><td>3.19 · 10<sup>-2</sup></td><td><b>6.76</b> · 10<sup>-2</sup></td><td><b>1.05</b> · 10<sup>-1</sup></td><td><b>2.32</b> · 10<sup>-1</sup></td><td><b>6.34</b> · 10<sup>-2</sup></td></tr><tr><td colspan="8" style="text-align:center; font-weight:bold;">5-step Autoregressive Evaluation</td></tr><tr><td rowspan="2">Base FNO</td><td>1×</td><td>4.54 · 10<sup>-1</sup></td><td>1.05 · 10<sup>-1</sup></td><td>1.96 · 10<sup>-1</sup></td><td>2.44 · 10<sup>-1</sup></td><td>7.03 · 10<sup>-1</sup></td><td>1.68 · 10<sup>-1</sup></td></tr><tr><td>2×</td><td>4.4 · 10<sup>-1</sup></td><td>9.17 · 10<sup>-2</sup></td><td>1.65 · 10<sup>-1</sup></td><td>1.55 · 10<sup>-1</sup></td><td>1.13 · 10<sup>0</sup></td><td>1.95 · 10<sup>-1</sup></td></tr><tr><td rowspan="2">NO-LIDK*</td><td>1×</td><td>4.32 · 10<sup>-1</sup></td><td>1.05 · 10<sup>-1</sup></td><td>2.03 · 10<sup>-1</sup></td><td>2.27 · 10<sup>-1</sup></td><td>8.58 · 10<sup>-1</sup></td><td>1.86 · 10<sup>-1</sup></td></tr><tr><td>2×</td><td>4.65 · 10<sup>-1</sup></td><td>9.89 · 10<sup>-2</sup></td><td>1.97 · 10<sup>-1</sup></td><td>1.58 · 10<sup>-1</sup></td><td>1.28 · 10<sup>0</sup></td><td>2.59 · 10<sup>-1</sup></td></tr><tr><td rowspan="2">NO-LIDK†</td><td>1×</td><td>3.85 · 10<sup>-1</sup></td><td>9.15 · 10<sup>-2</sup></td><td>1.69 · 10<sup>-1</sup></td><td>2.16 · 10<sup>-1</sup></td><td>3.75 · 10<sup>-1</sup></td><td>1.06 · 10<sup>-1</sup></td></tr><tr><td>2×</td><td>7.58 · 10<sup>-1</sup></td><td>2.59 · 10<sup>-1</sup></td><td>5.06 · 10<sup>-1</sup></td><td>2.11 · 10<sup>-1</sup></td><td><b>4.38</b> · 10<sup>-1</sup></td><td>1.78 · 10<sup>-1</sup></td></tr><tr><td rowspan="2">LOGLO-FNO</td><td>1×</td><td><b>3.78</b> · 10<sup>-1</sup></td><td><b>8.23</b> · 10<sup>-2</sup></td><td><b>1.55</b> · 10<sup>-1</sup></td><td><b>2.02</b> · 10<sup>-1</sup></td><td><b>3.56</b> · 10<sup>-1</sup></td><td><b>9.46</b> · 10<sup>-2</sup></td></tr><tr><td>2×</td><td><b>4.29</b> · 10<sup>-1</sup></td><td><b>7.95</b> · 10<sup>-2</sup></td><td><b>1.47</b> · 10<sup>-1</sup></td><td><b>1.51</b> · 10<sup>-1</sup></td><td><b>4.39</b> · 10<sup>-1</sup></td><td><b>1.4</b> · 10<sup>-1</sup></td></tr></tbody></table>

We provide the spatial ZSSR results of baseline FNO, NO-LIDK variants, and our proposed LOGLO-FNO on the test set of the 2D Kolmogorov Flow dataset (Li et al., 2022a) in Table 18. The original spatial resolution of the dataset is 128 × 128. Therefore, we perform an even rate 2× subsampling to obtain training data at a resolution of 64 × 64, denoted as 1× in the table. Naturally, the ZSSR task is then to evaluate the capability of the models for the next higher resolution of 128 × 128, for which we have the ground truth. We observe that baseline FNO exhibits superior spatial ZSSR capabilities, except on the spectral metrics (MELR & WLR), on which it incurs degradation. LOGLO-FNO, on the other hand, incurs degradation in spatial and spectral errors. However, it exhibits superior improvements on the frequency domain errors, outperforming base FNO on all metrics, but fRMSE (Low). Most importantly, we find that LOGLO-FNO is better than the NO-LIDK variants on all metrics. Subsequently, we perform ZSSR evaluation on the 5-step autoregressive rollouts, and find a similar trend.
<table><caption>Table 19: Spatial ZSSR evaluation of LOGLO-FNO on the test set of CNS Turb 2D dataset (Takamoto et al., 2022). Base FNO uses a spectral filter size of 40 and 65 hidden channels. LOGLO-FNO uses 40 and (16, 9) Fourier modes in the global and local branches, respectively, whereas the width is set to 65. NO-LIDK* uses 40 modes and 65 hidden channels and a radius cutoff value of 0.00390625. NO-LIDK* denotes using only localized integral kernel and NO-LIDK† means employing both disco and diff layers. Training spatial resolution is (256 × 256), being noted as 1×, whereas the ZSSR test resolution is (512 × 512) and denoted as 2×. We boldface the lowest errors and underline the value when the 2nd best model is LOGLO-FNO.</caption><thead><tr><th>Model</th><th>Resolution</th><th>nRMSE (↓)</th><th>fRMSE(L)</th><th>fRMSE(M)</th><th>fRMSE(H)</th><th>MELR (↓)</th><th>WLR (↓)</th></tr></thead><tbody><tr><td colspan="8" style="text-align: center; font-weight: bold;">1-step Evaluation</td></tr><tr><td rowspan="2">Base FNO</td><td>1x</td><td>4.82 · 10<sup>-2</sup></td><td>1.28 · 10<sup>-3</sup></td><td>1.78 · 10<sup>-3</sup></td><td>1.57 · 10<sup>-3</sup></td><td>2.78 · 10<sup>-1</sup></td><td>7.8 · 10<sup>-3</sup></td></tr><tr><td>2x</td><td>4.81 · 10<sup>-2</sup></td><td>1.26 · 10<sup>-3</sup></td><td>1.77 · 10<sup>-3</sup></td><td>9.24 · 10<sup>-4</sup></td><td>5.5 · 10<sup>-1</sup></td><td>7.85 · 10<sup>-3</sup></td></tr><tr><td rowspan="2">NO-LIDK*</td><td>1x</td><td>6.4 · 10<sup>-2</sup></td><td>1.5 · 10<sup>-3</sup></td><td>2.14 · 10<sup>-3</sup></td><td>1.98 · 10<sup>-3</sup></td><td>2.98 · 10<sup>-1</sup></td><td>1.31 · 10<sup>-2</sup></td></tr><tr><td>2x</td><td>7.61 · 10<sup>-2</sup></td><td>7.73 · 10<sup>-3</sup></td><td>2.92 · 10<sup>-3</sup></td><td>1.17 · 10<sup>-3</sup></td><td>5.29 · 10<sup>-1</sup></td><td>2.45 · 10<sup>-2</sup></td></tr><tr><td rowspan="2">NO-LIDK†</td><td>1x</td><td>5.57 · 10<sup>-2</sup></td><td>1.37 · 10<sup>-3</sup></td><td>1.89 · 10<sup>-3</sup></td><td>1.77 · 10<sup>-3</sup></td><td>2.08 · 10<sup>-1</sup></td><td>9.54 · 10<sup>-3</sup></td></tr><tr><td>2x</td><td>8.66 · 10<sup>-2</sup></td><td>4.8 · 10<sup>-3</sup></td><td>2.67 · 10<sup>-3</sup></td><td>1.93 · 10<sup>-3</sup></td><td>9.0 · 10<sup>-1</sup></td><td>1.21 · 10<sup>-2</sup></td></tr><tr><td rowspan="2">LOGLO-FNO</td><td>1x</td><td><b>4.34</b> · 10<sup>-2</sup></td><td><b>9.74</b> · 10<sup>-4</sup></td><td><b>1.17</b> · 10<sup>-3</sup></td><td><b>1.41</b> · 10<sup>-3</sup></td><td><b>1.83</b> · 10<sup>-1</sup></td><td><b>5.93</b> · 10<sup>-3</sup></td></tr><tr><td>2x</td><td><b>4.54</b> · 10<sup>-2</sup></td><td><b>1.06</b> · 10<sup>-3</sup></td><td><b>1.27</b> · 10<sup>-3</sup></td><td><b>8.91</b> · 10<sup>-4</sup></td><td><b>3.56</b> · 10<sup>-1</sup></td><td><b>7.75</b> · 10<sup>-3</sup></td></tr><tr><td colspan="8" style="text-align: center; font-weight: bold;">5-step Autoregressive Evaluation</td></tr><tr><td rowspan="2">Base FNO</td><td>1x</td><td>7.78 · 10<sup>-2</sup></td><td>3.76 · 10<sup>-3</sup></td><td>5.37 · 10<sup>-3</sup></td><td>2.4 · 10<sup>-3</sup></td><td>2.22 · 10<sup>-1</sup></td><td>9.36 · 10<sup>-3</sup></td></tr><tr><td>2x</td><td><b>7.06</b> · 10<sup>-2</sup></td><td>3.31 · 10<sup>-3</sup></td><td>4.77 · 10<sup>-3</sup></td><td><b>1.37</b> · 10<sup>-3</sup></td><td>5.12 · 10<sup>-1</sup></td><td><b>8.1</b> · 10<sup>-3</sup></td></tr><tr><td rowspan="2">NO-LIDK*</td><td>1x</td><td>8.55 · 10<sup>-2</sup></td><td>4.14 · 10<sup>-3</sup></td><td>5.92 · 10<sup>-3</sup></td><td>2.62 · 10<sup>-3</sup></td><td>2.06 · 10<sup>-1</sup></td><td>1.15 · 10<sup>-2</sup></td></tr><tr><td>2x</td><td>1.35 · 10<sup>-1</sup></td><td>1.5 · 10<sup>-2</sup></td><td>1.1 · 10<sup>-2</sup></td><td>2.11 · 10<sup>-3</sup></td><td>4.25 · 10<sup>-1</sup></td><td>3.48 · 10<sup>-2</sup></td></tr><tr><td rowspan="2">NO-LIDK†</td><td>1x</td><td>7.91 · 10<sup>-2</sup></td><td>3.82 · 10<sup>-3</sup></td><td>5.38 · 10<sup>-3</sup></td><td>2.47 · 10<sup>-3</sup></td><td>1.45 · 10<sup>-1</sup></td><td>1.01 · 10<sup>-2</sup></td></tr><tr><td>2x</td><td>3.9 · 10<sup>0</sup></td><td>4.17 · 10<sup>-1</sup></td><td>1.41 · 10<sup>-1</sup></td><td>7.21 · 10<sup>-2</sup></td><td>2.36 · 10<sup>0</sup></td><td>5.25 · 10<sup>-1</sup></td></tr><tr><td rowspan="2">LOGLO-FNO</td><td>1x</td><td><b>6.72</b> · 10<sup>-2</sup></td><td><b>2.8</b> · 10<sup>-3</sup></td><td><b>4.0</b> · 10<sup>-3</sup></td><td><b>2.14</b> · 10<sup>-3</sup></td><td><b>1.31</b> · 10<sup>-1</sup></td><td><b>6.78</b> · 10<sup>-3</sup></td></tr><tr><td>2x</td><td><b>7.31</b> · 10<sup>-2</sup></td><td><b>3.2</b> · 10<sup>-3</sup></td><td><b>4.36</b> · 10<sup>-3</sup></td><td><b>1.43</b> · 10<sup>-3</sup></td><td><b>3.13</b> · 10<sup>-1</sup></td><td><b>8.36</b> · 10<sup>-3</sup></td></tr><tr><td colspan="8" style="text-align: center; font-weight: bold;">20-step (entire trajectory) Autoregressive Evaluation</td></tr><tr><td rowspan="2">Base FNO</td><td>1x</td><td>1.81 · 10<sup>-1</sup></td><td>1.25 · 10<sup>-2</sup></td><td>1.44 · 10<sup>-2</sup></td><td>3.26 · 10<sup>-3</sup></td><td>2.8 · 10<sup>-1</sup></td><td>3.22 · 10<sup>-2</sup></td></tr><tr><td>2x</td><td>1.82 · 10<sup>-1</sup></td><td>1.26 · 10<sup>-2</sup></td><td>1.45 · 10<sup>-2</sup></td><td><b>1.78</b> · 10<sup>-3</sup></td><td>5.52 · 10<sup>-1</sup></td><td>3.24 · 10<sup>-2</sup></td></tr><tr><td rowspan="2">NO-LIDK*</td><td>1x</td><td>1.97 · 10<sup>-1</sup></td><td>1.44 · 10<sup>-2</sup></td><td>1.6 · 10<sup>-2</sup></td><td>3.36 · 10<sup>-3</sup></td><td>2.95 · 10<sup>-1</sup></td><td>3.94 · 10<sup>-2</sup></td></tr><tr><td>2x</td><td>3.72 · 10<sup>-1</sup></td><td>4.63 · 10<sup>-2</sup></td><td>3.16 · 10<sup>-2</sup></td><td>2.39 · 10<sup>-3</sup></td><td>3.66 · 10<sup>-1</sup></td><td>1.09 · 10<sup>-1</sup></td></tr><tr><td rowspan="2">NO-LIDK†</td><td>1x</td><td>1.87 · 10<sup>-1</sup></td><td>1.35 · 10<sup>-2</sup></td><td>1.48 · 10<sup>-2</sup></td><td>3.37 · 10<sup>-3</sup></td><td>2.1 · 10<sup>-1</sup></td><td>3.7 · 10<sup>-2</sup></td></tr><tr><td>2x</td><td>1.04 · 10<sup>11</sup></td><td>1.18 · 10<sup>10</sup></td><td>5.64 · 10<sup>8</sup></td><td>1.77 · 10<sup>9</sup></td><td>1.33 · 10<sup>1</sup></td><td>9.99 · 10<sup>0</sup></td></tr><tr><td rowspan="2">LOGLO-FNO</td><td>1x</td><td><b>1.62</b> · 10<sup>-1</sup></td><td><b>1.07</b> · 10<sup>-2</sup></td><td><b>1.26</b> · 10<sup>-2</sup></td><td><b>3.09</b> · 10<sup>-3</sup></td><td><b>1.87</b> · 10<sup>-1</sup></td><td><b>2.7</b> · 10<sup>-2</sup></td></tr><tr><td>2x</td><td><b>1.72</b> · 10<sup>-1</sup></td><td><b>1.16</b> · 10<sup>-2</sup></td><td><b>1.34</b> · 10<sup>-2</sup></td><td><b>1.79</b> · 10<sup>-3</sup></td><td><b>3.49</b> · 10<sup>-1</sup></td><td><b>3.02</b> · 10<sup>-2</sup></td></tr></tbody></table>
# M Training and Inference Time Comparison of LOGLO-FNO

## M.1 Comparison with Numerical Solver

The inference time of LOGLO-FNO to generate predictions on the Turbulent Radiative Layer 3D dataset is provided in Table 20 and compared against the CPU-only numerical solver Athena++ used to generate the original simulations of the TRL3D dataset. However, in order to establish a fair comparison with the numerical solver, the training time of the neural surrogate (LOGLO-FNO), which is about 12 hours for 50 epochs on TRL3D, should also be considered.

<table><caption>Table 20: Comparison of runtimes to generate all trajectories at the full resolution for the Turbulent Radiative Layer 3D dataset.</caption><thead><tr><th>Solver Type</th><th>Hardware</th><th>Runtime</th></tr></thead><tbody><tr><td>Numerical Solver: Athena++<br>(Stone et al., 2020; Ohana et al., 2024)</td><td>CPU (128 cores)</td><td>34,560 CPU-h</td></tr><tr><td>Neural Surrogate: LOGLO-FNO</td><td>one H100 GPU</td><td>38.143767 mins</td></tr></tbody></table>

<table><caption>Table 21: Scaling factor of runtimes to generate all trajectories at training and higher (2×) spatial resolutions for the Turbulent Radiative Mixing Layer 3D dataset.</caption><thead><tr><th>Metric</th><th>Data Split</th><th>Spatial Resolution<br>(64×64×128)</th><th>Spatial Resolution<br>(128×128×256)</th><th>Scaling Factor</th></tr></thead><tbody><tr><td>Total Grid Points</td><td>×</td><td>524,288</td><td>4,194,304</td><td>8x</td></tr><tr><td>Runtime (9 Trajectories)</td><td>Test</td><td>31,430.94 ± 0.80 ms<br>(~31.4 sec)</td><td>228,862.60 ± 0.81 ms<br>(~3.8 min)</td><td>~7.28x</td></tr><tr><td>Runtime (90 Trajectories)</td><td>Full</td><td>314,309.4 ± 8.0 ms<br>(~5.2 min)</td><td>2,288,626.0 ± 8.1 ms<br>(~38.1 min)</td><td>~7.28x</td></tr></tbody></table>

We measure the runtimes by generating the trajectory predictions only for the *test set*, containing 9 trajectories, each of which has 101 timesteps (row 2 in Table 21). The runtime consumed by the numerical solver Athena++ is taken from Ohana et al. (2024)<fnref n="1" />. The runtime for the full dataset containing 90 trajectories (row 3 in Table 21) is obtained by naive scaling (i.e., multiplying by 10) of the GPU runtime for the 9 trajectories. During inference, we use a batch size of 1 and report the mean runtime and standard deviation over 10 iterations for both spatial resolutions listed in Table 21.

## M.2 Training & Inference Time and GPU Memory Usage Comparison with Base FNO and NO-LIDK

In this part, we provide the training and inference times as well as the GPU memory usage of LOGLO-FNO and compare them with FNO and NO-LIDK variants of baseline models since these are the most related model architectures for LOGLO-FNO. We note that all experiments, configured through the Slurm workload manager, are conducted on a single H100 NVIDIA graphics card with 96 GB of memory. The benchmark problem we consider is the 2D Kolmogorov Flow (Li et al., 2022a) and 1-step training outlined in §4.0.1.

In order to establish a complete picture of the trade-off among these models in terms of memory usage and total epoch times, we list the *training times* and GPU memory usage for the forward and backward passes in Table 22. We observe that baseline FNO (F) wins all other models, both in terms of GPU memory consumed and time needed for the completion of the full training cycle of 136 epochs. NO-LIDK (L), employing a differential kernel as a parallel branch, occupies the second place. LOGLO-FNO (F+L) comes third in training time, while it requires slightly more memory than NO-LIDK*, which uses a local integral kernel, takes fourth

<footnote n="1">¹https://polymathic-ai.org/the_well/datasets/turbulent_radiative_layers_3D</footnote>
place when comparing training time. In last place is the NO-LIDK variant, which utilizes both kernels as parallel branches in addition to the global Fourier branch.

<table><caption>Table 22: Training times and GPU memory usage for the forward and backward passes during the 1-step training for a total of 136 epochs on the Kolmogorov Flow 2D training dataset. The reported values are the average over 4 training runs on the training set. NO-LIDK*, NO-LIDK°, and NO-LIDK† represent the variants trained with the local integral (DISCO), differential kernel, and both layers as additional parallel branches, in that corresponding order.</caption><thead><tr><th>Model</th><th>Training Time</th><th>Slowdown wrt. FNO</th><th>GPU Mem. (GB)</th><th>Mem. Fact. vs. FNO</th><th>Batch Size</th><th>Spatial Res.</th><th>Fourier Modes</th><th>Hidden Channel</th><th>P<sub>size</sub></th><th>Radius Cutoff</th></tr></thead><tbody><tr><td>Base FNO</td><td>1h 36m 47s ± 8.35s</td><td>ref.</td><td>33.90 ± 0.81</td><td>ref.</td><td>256</td><td>128x128</td><td>40</td><td>65</td><td>✗</td><td>✗</td></tr><tr><td>LOGLO-FNO</td><td>5h 19m 40s ± 16.28s</td><td>3.30x</td><td>76.02 ± 2.10</td><td>2.24x</td><td>256</td><td>128x128</td><td>40</td><td>65</td><td>16 x 16</td><td>✗</td></tr><tr><td>NO-LIDK*</td><td>7h 23m 30s ± 16.59s</td><td>4.58x</td><td>69.33 ± 2.02</td><td>2.05x</td><td>256</td><td>128x128</td><td>20</td><td>129</td><td>✗</td><td>0.05π</td></tr><tr><td>NO-LIDK°</td><td>2h 52m 54s ± 7.55s</td><td>1.79x</td><td>35.63 ± 1.12</td><td>1.05x</td><td>256</td><td>128x128</td><td>40</td><td>65</td><td>✗</td><td>✗</td></tr><tr><td>NO-LIDK†</td><td>7h 31m 11s ± 13.54s</td><td>4.66x</td><td>78.75 ± 4.63</td><td>2.32x</td><td>256</td><td>128x128</td><td>20</td><td>127</td><td>✗</td><td>0.05π</td></tr></tbody></table>

Subsequently, we compute the inference times and GPU memory usage for the 1-step predictions and list them in Table 23, with the provided values averaged over 100 repetitions.

<table><caption>Table 23: Inference times and GPU memory usage for the forward pass during 1-step evaluation on the Kolmogorov Flow 2D test dataset. The reported values are the average over 100 iterations on the test set. NO-LIDK*, NO-LIDK°, and NO-LIDK† represent the variants trained with the local integral (DISCO), differential kernel, and both layers as additional parallel branches, in that corresponding order.</caption><thead><tr><th>Model</th><th>Inference Time (ms)</th><th>Slowdown wrt. FNO</th><th>GPU Mem. (GB)</th><th>Batch Size</th><th>Spatial Res.</th><th>Fourier Modes</th><th>Hidden Channel</th><th>P<sub>size</sub></th><th>Radius Cutoff</th></tr></thead><tbody><tr><td>Base FNO</td><td>1471.41 ± 0.06</td><td>ref.</td><td>10.74 ± 0.0</td><td>256</td><td>128x128</td><td>40</td><td>65</td><td>✗</td><td>✗</td></tr><tr><td>LOGLO-FNO</td><td>4899.53 ± 0.15</td><td>3.33x</td><td>18.31 ± 0.0</td><td>256</td><td>128x128</td><td>40</td><td>65</td><td>16 x 16</td><td>✗</td></tr><tr><td>NO-LIDK*</td><td>7601.22 ± 7.17</td><td>5.17x</td><td>19.63 ± 0.0</td><td>256</td><td>128x128</td><td>20</td><td>129</td><td>✗</td><td>0.05π</td></tr><tr><td>NO-LIDK°</td><td>2477.68 ± 0.14</td><td>1.68x</td><td>10.19 ± 0.0</td><td>256</td><td>128x128</td><td>40</td><td>65</td><td>✗</td><td>✗</td></tr><tr><td>NO-LIDK†</td><td>7096.81 ± 4.34</td><td>4.82x</td><td>21.43 ± 0.0</td><td>256</td><td>128x128</td><td>20</td><td>127</td><td>✗</td><td>0.05π</td></tr></tbody></table>

We observe a similar trend with baseline FNO being the fastest, however, consuming slightly higher GPU memory than NO-LIDK°, which is about 1.7x slower. In the third place is our LOGLO-FNO model, incurring a 3.3x slowdown factor. Pertinent to our model is NO-LIDK*, and it is about 1.6x slower while also incurring a slightly higher memory consumption than LOGLO-FNO under the considered hyperparameter configurations, following Liu-Schiaffini et al. (2024). Finally, the NO-LIDK† variant, consisting of both branches, incurs the highest GPU memory, although it is slightly faster than the NO-LIDK* variant with only the local integral kernel branch as an additional pathway.
## N Confidence Analysis of LOGLO-FNO and Baselines

In this section, we report the mean and standard deviation values for the base FNO and NO-LIDK variants, comparing them with our proposed LOGLO-FNO model. We repeat the 1-step training setup discussed in §4.0.1 on the 2D Kolmogorov Flow dataset four times with different random seeds, and evaluate the 1-step as well as the 5-step autoregressive rollouts. The plot in Figure 39 shows the mean and 95% confidence interval values of fRMSE and WLR errors, with the hatched and solid bars visualizing the 1-step and 5-step autoregressive rollout errors in that order.

<figure><img src="image_43.png" /><figcaption>Figure 39: Confidence analysis of baseline FNO, NO-LIDK (Liu-Schiaffini et al., 2024), and proposed LOGLO-FNO for the 1-step and 5-step autoregressive evaluation of fRMSE and WLR (Wan et al., 2023) metrics on the test set of 2D Kolmogorov Flow (Li et al., 2022b). NO-LIDK\* and NO-LIDK\° represent the variants trained with the local integral (DISCO) and differential kernel layers, respectively.</figcaption></figure>

We observe that LOGLO-FNO consistently outperforms baseline FNO and is significantly better than NO-LIDK\*, utilizing the local integral kernel, across all metrics. Particularly noteworthy are the improvements we observe in the rollout scenarios.

In a similar vein, in Figure 40, we visualize the mean and 95% confidence interval values of nRMSE and MELR errors, whereas Figure 41 establishes a comparison of the plain and boundary RMSE values.

We find that while LOGLO-FNO is the best model on the nRMSE metric, NO-LIDK\° employing a differential kernel layer outperforms all models, reinforcing the results found in Table 8. However, we note that the results on NO-LIDK\° are provided only for the sake of completeness, and LOGLO-FNO should be compared with NO-LIDK\* when assessing its effectiveness.

Liu-Schiaffini et al. (2024) find the differential kernel layer to be effective in reducing the boundary error for non-periodic boundary conditions. Additionally, in our study on Kolmogorov Flow 2D, it has been found to be effective for periodic boundary conditions as well, enforced through padding. However, this advantage is not maintained in rollouts (see Figure 41).
<figure><img src="image_44.png" /><figcaption>Figure 40: Confidence analysis of baseline FNO, NO-LIDK (Liu-Schiaffini et al., 2024), and proposed LOGLO-FNO for the 1-step and 5-step autoregressive evaluation of nRMSE and MELR (Wan et al., 2023) metrics on the test set of 2D Kolmogorov Flow (Li et al., 2022b). NO-LIDK\* and NO-LIDK\circ represent the variants trained with the local integral (DISCO) and differential kernel layers, respectively.</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>Figure 41: Confidence analysis of baseline FNO, NO-LIDK (Liu-Schiaffini et al., 2024), and proposed LOGLO-FNO for the 1-step and 5-step autoregressive evaluation of xRMSE (plain and boundary) metrics on the test set of 2D Kolmogorov Flow (Li et al., 2022b). NO-LIDK\* and NO-LIDK\circ represent the variants trained with the local integral (DISCO) and differential kernel layers, respectively.</figcaption></figure>
# O Model Hyperparameters

## O.1 Modern U-Net Hyperparameters

The Table 24 below presents the hyperparameters for the modern version of U-Net baseline we use from the PDEArena (Gupta & Brandstetter, 2023) benchmark codebase.

<table><caption>Table 24: Model hyperparameters for the modern U-Net (Gupta & Brandstetter, 2023) baseline.</caption><thead><tr><th>PDE Problem</th><th>Channel Multiplier</th><th>Learning Rate (LR)</th><th>LR Sched.</th><th>Sched. Steps</th><th>Sched. Gamma</th><th>Train Type</th><th>Train Loss</th><th>Total Epochs</th></tr></thead><tbody><tr><td>2D Kolmogorov Flow</td><td>(1, 2, 2, 3, 4)</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D CNS Turb</td><td>(1, 2, 2, 3, 4)</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr></tbody></table>

## O.2 Base FNO Hyperparameters

The Table 25 below lists the hyperparameters for training the FNO baseline model on the two-dimensional and three-dimensional PDE problems considered in our experiments. Note that we borrow the hyperparameters from Liu-Schiaffini et al. (2024) for the Kolmogorov Flow case and Takamoto et al. (2022) for the Diffusion-Reaction 2D PDE.

<table><caption>Table 25: Model hyperparameters for FNO baseline.</caption><thead><tr><th>PDE Problem</th><th>Fourier Modes</th><th>Channel Dimens.</th><th>Learning Rate (LR)</th><th>LR Scheduler</th><th>Sched. Steps</th><th>Sched. Gamma</th><th>Train Type</th><th>Train Loss</th><th>Total Epochs</th></tr></thead><tbody><tr><td>2D Diffusion-Reaction</td><td>12</td><td>20</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>100</td><td>0.5</td><td>AR</td><td>MSE</td><td>500</td></tr><tr><td>2D Kolmogorov Flow</td><td>40</td><td>65</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D Wave-Gauss</td><td>20</td><td>32</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D CNS Turb</td><td>40</td><td>65</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D Comp. Euler RP</td><td>40</td><td>65</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>Turbulent Radiative Layer 3D</td><td>12</td><td>48</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>10</td><td>0.5</td><td>1-step</td><td>MSE</td><td>50</td></tr></tbody></table>

## O.3 F-FNO Hyperparameters

Table 26 provides the hyperparameters for the F-FNO baseline model from Tran et al. (2023). As with the baseline FNO model, we use four Fourier layers.

<table><caption>Table 26: Model hyperparameters for F-FNO (Tran et al., 2023) baseline.</caption><thead><tr><th>PDE Problem</th><th>Fourier Modes</th><th>Channel Dimens.</th><th>Learning Rate (LR)</th><th>LR Scheduler</th><th>Warmup Steps</th><th>Train Type</th><th>Train Loss</th><th>Total Epochs</th></tr></thead><tbody><tr><td>2D Diffusion-Reaction</td><td>16</td><td>32</td><td>6.0 ·e<sup>-4</sup></td><td>Cosine Warmup</td><td>200</td><td>AR</td><td>MSE</td><td>500</td></tr><tr><td>2D Kolmogorov Flow</td><td>40</td><td>65</td><td>1.0 ·e<sup>-3</sup></td><td>Cosine Warmup</td><td>60</td><td>1-step</td><td>MSE</td><td>136</td></tr></tbody></table>

## O.4 LSM Hyperparameters

Tables 27 provides the hyperparameters used for training the Latent Spectral Model (Wu et al., 2023) baseline on the Kolmogorov Flow 2D, CNS Turb 2D, Wave-Gauss 2D, and Diffusion-Reaction 2D datasets, whereas Table 28 lists the model hyperparameters used for the LSM architecture configuration.

We found the model to be very sensitive to the learning rate in the full autoregressive training setup and, hence, we use a low learning rate of $5.0 \cdot e^{-5}$ in addition to employing gradient clipping with a max norm of 5.0.
<table><caption>Table 27: Training settings for the LSM (Wu et al., 2023) baseline model.</caption><thead><tr><th>PDE Problem</th><th>Learning Rate (LR)</th><th>LR Scheduler</th><th>Sched. Steps</th><th>Sched. Gamma</th><th>Train Type</th><th>Train Loss</th><th>Total Epochs</th></tr></thead><tbody><tr><td>2D Diffusion-Reaction</td><td>5.0 ·e<sup>-5</sup></td><td>StepLR</td><td>100</td><td>0.5</td><td>AR</td><td>MSE</td><td>500</td></tr><tr><td>2D Kolmogorov Flow</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D CNS Turb</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D Wave-Gauss</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>3D Turbulent Radiative Layer</td><td>1.0 ·e<sup>-3</sup></td><td>StepLR</td><td>10</td><td>0.5</td><td>1-step</td><td>MSE</td><td>50</td></tr></tbody></table>

<table><caption>Table 28: Further Model hyperparameters for LSM (Wu et al., 2023) baseline.</caption><thead><tr><th>PDE Problem</th><th>Num. Tokens</th><th>Num. Basis</th><th>Channel Dimens. (init.)</th><th>Downsample Ratio</th><th>Num. Scales</th><th>Patch Size</th><th>Interpol. Type</th></tr></thead><tbody><tr><td>2D Diffusion-Reaction</td><td>4</td><td>8</td><td>32</td><td>5/10</td><td>5</td><td>(4,4)</td><td>bilinear</td></tr><tr><td>2D Kolmogorov Flow</td><td>4</td><td>16</td><td>64</td><td>5/10</td><td>5</td><td>(4,4)</td><td>bilinear</td></tr><tr><td>2D CNS Turb</td><td>4</td><td>16</td><td>64</td><td>5/10</td><td>5</td><td>(4,4)</td><td>bilinear</td></tr><tr><td>2D Wave-Gauss</td><td>4</td><td>8</td><td>32</td><td>5/10</td><td>5</td><td>(4,4)</td><td>bilinear</td></tr><tr><td>3D Turbulent Radiative Layer</td><td>4</td><td>16</td><td>64</td><td>5/10</td><td>5</td><td>(4,4,8)</td><td>trilinear</td></tr></tbody></table>

Channel dimensions (init.) represent the number of channels used for the incoming full spatial resolution, including which the remaining four spatial scales and their respective channels follow.

## 0.5 Transolver Hyperparameters

Tables 29 provides the hyperparameters used for training the Transolver Model (Wu et al., 2024) baseline on the Kolmogorov Flow 2D, CNS Turb 2D, Wave-Gauss 2D, and Diffusion-Reaction 2D datasets, whereas Table 30 lists the model hyperparameters used for the Transolver model architecture configuration.

<table><caption>Table 29: Training settings for the Transolver (Wu et al., 2024) baseline model.</caption><thead><tr><th>PDE Problem</th><th>Optimizer</th><th>Learning Rate (LR)</th><th>LR Scheduler</th><th>Weight Decay</th><th>Train Type</th><th>Train Loss</th><th>Total Epochs</th></tr></thead><tbody><tr><td>2D Diffusion-Reaction</td><td>AdamW</td><td>4.0 ·e<sup>-4</sup></td><td>OneCycleLR</td><td>1.0 ·e<sup>-4</sup></td><td>AR</td><td>MSE</td><td>500</td></tr><tr><td>2D Kolmogorov Flow</td><td>AdamW</td><td>6.0 ·e<sup>-3</sup></td><td>OneCycleLR</td><td>1.0 ·e<sup>-4</sup></td><td>1-step</td><td>MSE</td><td>500</td></tr><tr><td>2D CNS Turb</td><td>AdamW</td><td>6.0 ·e<sup>-3</sup></td><td>OneCycleLR</td><td>1.0 ·e<sup>-4</sup></td><td>1-step</td><td>MSE</td><td>500</td></tr><tr><td>2D Wave-Gauss</td><td>AdamW</td><td>6.0 ·e<sup>-3</sup></td><td>OneCycleLR</td><td>1.0 ·e<sup>-4</sup></td><td>1-step</td><td>MSE</td><td>500</td></tr></tbody></table>

<table><caption>Table 30: Model architecture hyperparameters for Transolver (Wu et al., 2024) baseline.</caption><thead><tr><th>PDE Problem</th><th>Num. Layers</th><th>Num. Heads</th><th>Num. Slices</th><th>Hidden Channels</th><th>Activation Function</th><th>MLP Ratio</th><th>Apply Dropout</th></tr></thead><tbody><tr><td>2D Diffusion-Reaction</td><td>4</td><td>8</td><td>32</td><td>64</td><td>ReLU</td><td>1</td><td>False</td></tr><tr><td>2D Kolmogorov Flow</td><td>9</td><td>8</td><td>64</td><td>128</td><td>ReLU</td><td>1</td><td>False</td></tr><tr><td>2D CNS Turb</td><td>9</td><td>8</td><td>64</td><td>128</td><td>ReLU</td><td>1</td><td>False</td></tr><tr><td>2D Wave-Gauss</td><td>9</td><td>8</td><td>64</td><td>32</td><td>ReLU</td><td>1</td><td>False</td></tr></tbody></table>
## O.6 U-FNO Hyperparameters

Table 31 below lists the hyperparameters for the U-FNO baseline model from Wen et al. (2022).

<table><caption>Table 31: Model hyperparameters for U-FNO (Wen et al., 2022) baseline.</caption><thead><tr><th>PDE Problem</th><th>Fourier Modes</th><th>Channel Dimens.</th><th>Learning Rate (LR)</th><th>LR Scheduler</th><th>Sched. Steps</th><th>Sched. Gamma</th><th>Train Type</th><th>Train Loss</th><th>Total Epochs</th></tr></thead><tbody><tr><td>2D Diffusion-Reaction</td><td>10</td><td>19</td><td>5.0 ·e⁻⁴</td><td>StepLR</td><td>10</td><td>0.9</td><td>AR</td><td>MSE</td><td>500</td></tr><tr><td>2D Kolmogorov Flow</td><td>36</td><td>48</td><td>1.0 ·e⁻³</td><td>StepLR</td><td>2</td><td>0.9</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D CNS Turb</td><td>36</td><td>48</td><td>5.0 ·e⁻⁴</td><td>StepLR</td><td>10</td><td>0.9</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D Wave-Gauss</td><td>16</td><td>32</td><td>5.0 ·e⁻⁴</td><td>StepLR</td><td>10</td><td>0.9</td><td>1-step</td><td>MSE</td><td>136</td></tr></tbody></table>

## O.7 NO-LIDK Hyperparameters

Table 32 lists the hyperparameters for different configurations of the Neural Operators with Localized Integral and Differential Kernels (NO-LIDK) model from Liu-Schiaffini et al. (2024). For the Kolmogorov Flow 2D data, following the authors' training setup (see Table 4 in Liu-Schiaffini et al. (2024)), we use 20 modes and a hidden channel dimension of 127 for the model employing both local integral and differential kernel layers (NO-LIDK†), 20 modes and a hidden channel dimension of 129 for the model using only the local integral kernel layer (NO-LIDK*), and 40 modes and a hidden channel dimension of 65 for the model with just the differential kernel layer (NO-LIDK°) as parallel layers in addition to the global spectral convolution layer of base FNO. All these variants use 4 Fourier layers in total.

<table><caption>Table 32: Model hyperparameters for NO-LIDK (Liu-Schiaffini et al., 2024) baseline.</caption><thead><tr><th>PDE Problem</th><th>Integral Kernel</th><th>Differential Kernel</th><th>Learning Rate (LR)</th><th>LR Sched.</th><th>Sched. Steps</th><th>Sched. Gamma</th><th>Train Type</th><th>Train Loss</th><th>Total Epochs</th></tr></thead><tbody><tr><td>2D Kolmogorov Flow</td><td>✓</td><td>✓</td><td>1.0 ·e⁻³</td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D Wave-Gauss</td><td>✓</td><td>✓</td><td>1.0 ·e⁻³</td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D Comp. Euler RP</td><td>✓</td><td>✓</td><td>1.0 ·e⁻³</td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>2D CNS Turb</td><td>✓</td><td>✓</td><td>1.0 ·e⁻³</td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE</td><td>136</td></tr><tr><td>Turbulent Radiative Layer 3D</td><td>✗</td><td>✓</td><td>6.0 ·e⁻³</td><td>OneCycleLR</td><td>10</td><td>✗</td><td>1-step</td><td>MSE</td><td>50</td></tr></tbody></table>

The local integral kernel for the disco convolution employs a radius cutoff value of 0.0078125, 0.0078125, and 0.00390625 for the Wave-Gauss 2D, Compressible Euler four-quadrant Riemann problem 2D, and CNS Turb 2D problems, respectively, following the heuristic suggested by Liu-Schiaffini et al. (2024). As in the case of the baseline FNO model, the number of spectral filters and hidden channels is set as 40 and 65, resp., for all three configurations of the trained models (i.e., disco layers – NO-LIDK*, differential layers – NO-LIDK°, and disco+differential layers – NO-LIDK†) for the CNS Turb 2D dataset, whereas we use 20 modes and 32 hidden channels for the Wave-Gauss 2D problem. We use the hyperparameter settings applied for 2D Kolmogorov Flow for the Compressible Euler four-quadrant Riemann problem 2D PDE.

For the Turbulent Radiative Layer 3D data, we use 16, 16, and 20 modes along the x, y, and z axes. The number of hidden channels is set as 96, and the model consists of 4 layers, as in the case of all 2D experiments.

## O.8 LOGLO-FNO Implementation and Hyperparameters

**Implementation.** Our implementation uses PyTorch v2 (Ansel et al., 2024), NumPy (Harris et al., 2020), and FNO base implementation adapted from the neural operator library (Kossaifi et al., 2024a).

**Hyperparameters.** The Kolmogorov Flow problem uses a batch size of 256 and a patch size of 16 × 16 (in the local branch) for the 1-step training strategy, whereas it is set as 8 × 8 for Diffusion-Reaction 2D due to memory limitations with fully autoregressive training on long trajectories of 91 timesteps. An experiment
employing data-parallel training using a patch size of 16 has not resulted in improved performance. Therefore, we set the patch size to 8 when extracting non-overlapping patches on the domain for the Diffusion-Reaction problem and report the results for this setting.

<table><caption>Table 33: Model hyperparameters for our proposed LOGLO-FNO in the main paper with simple summation (+) as the fusion method (Fig. 1) for the features from the HFP, global, and local branches.</caption><thead><tr><th>PDE Problem</th><th>Fourier Modes (Global)</th><th>Channel Dimens.</th><th>Learning Rate (LR)</th><th>LR Scheduler</th><th>Sched. Steps</th><th>Sched. Gamma</th><th>Train Type</th><th>Train Loss</th><th>Total Epochs</th></tr></thead><tbody><tr><td>2D Diffusion-Reaction</td><td>10</td><td>24</td><td>8.0 ·e⁻⁴</td><td>StepLR</td><td>100</td><td>0.5</td><td>AR</td><td>MSE + (mid&high) Freq.</td><td>500</td></tr><tr><td>2D Kolmogorov Flow</td><td>40</td><td>65</td><td>4.4 ·e⁻³</td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE + (mid&high) Freq.</td><td>136</td></tr><tr><td>2D Wave-Gauss</td><td>20</td><td>32</td><td>4.4 ·e⁻³</td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE + (mid&high) Freq.</td><td>136</td></tr><tr><td>2D Comp. Euler RP</td><td>40</td><td>65</td><td>4.4 ·e⁻³</td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE + (mid&high) Freq.</td><td>136</td></tr><tr><td>2D CNS Turb</td><td>40</td><td>65</td><td>4.4 ·e⁻³</td><td>StepLR</td><td>33</td><td>0.5</td><td>1-step</td><td>MSE + (mid&high) Freq.</td><td>136</td></tr><tr><td>Turbulent Radiative Layer 3D</td><td>12</td><td>52</td><td>4.0 ·e⁻³</td><td>StepLR</td><td>10</td><td>0.5</td><td>1-step</td><td>MSE + (mid&high) Freq.</td><td>50</td></tr></tbody></table>

All models use a batch size of 64, 256, and 256 for the CNS Turb 2D, Compressible Euler four-quadrant Riemann problem 2D, and Wave-Gauss 2D problems, respectively. LOGLO-FNO uses a patch size of 16 × 16 for the Compressible Euler four-quadrant Riemann problem 2D and Wave-Gauss 2D from Poseidon (Herde et al., 2024) and CNS Turb 2D dataset, which is provided by PDEBench (Takamoto et al., 2022).

The kernel size and stride have been set to 4 in the high-frequency extraction and propagation modules for all PDE problems. The local branch (e.g., with a patch size of 8 × 8 or 16 × 16) uses all available frequency modes. This would yield 16 and 9 Fourier modes for the spatial dimensions x and y, respectively, when a patch size of 16×16 is considered. Analogously, assuming an equal number of sampling points along all three spatial axes of a 3D spatial data, the local branch configured with a patch size of 16 × 16 × 16 would use all the 16, 16, and 9 Fourier modes for the spatial dimensions x, y, and z, respectively. Since Turbulent Radiative Layer 3D (Ohana et al., 2024) has 2x more observation points on the z-axis (i.e., 128 × 128 × 256), we use a proportionate patch size of 16 × 16 × 32, resulting in 16, 16, and 17 Fourier modes along the x, y, and z axes, respectively.
P Table of Notations and Mathematical Symbols

To serve as a handy guide, the following tabulation describes the symbols used within this manuscript.

$\mathbb{R}_+$ Set of positive real numbers. Specifically, $t \in (0, T]$

$\nabla$ Gradient or vector derivative operator ($\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \dots$)

$\Delta$ Laplacian ($\nabla^2$)

$A, U$ Banach spaces of functions

$\partial_t$ Partial derivative with respect to $t$

$\partial_x$ Partial derivative with respect to $x$

$\partial_y$ Partial derivative with respect to $y$

$u(x, t)$ Solution of a PDE at time $t$ for a given spatial coordinate $x$

$\mathbf{v}$ Velocity vector field ($\vec{x}, \vec{y}, \dots$)

$\mathbf{p}$ PDE parameter values, either a scalar or vector

$\mathcal{N}_{\theta}$ Neural network with learnable parameters ($\theta$)

$\mathcal{P}, \mathcal{Q}$ Lifting and projection layers, respectively

$\mathcal{K}_l, \mathcal{K}_g$ Local and global kernel integral operator, respectively

$\mathcal{L}$ Number of layers in the network

$\mathcal{C}$ Cost function

$u^0(x)$ Initial data of the PDE at time $t=0$

$\Omega$ Domain of the PDE under consideration

$\partial\Omega$ Boundary of the domain $\Omega$ under consideration

$N_t$ Total number of timesteps in the simulation

$N_x$ Total number of spatial points along the x-axis of the simulation

$N_y$ Total number of spatial points along the y-axis of the simulation

$N_z$ Total number of spatial points along the z-axis of the simulation

$N_c$ Total number of channels or fields in the simulation

$d_c$ channel dimension a.k.a. width of the network

$p_{size}$ Patch size or sub-domain size

$\nu$ Viscosity coefficient of Navier-Stokes equations

$Re$ Reynolds Number

$\rho$ Mass density in Navier-Stokes equations

$\eta$ Shear viscosity in Compressible Navier-Stokes equations

$\zeta$ Bulk viscosity in Compressible Navier-Stokes equations

$\epsilon$ Internal energy in Compressible Navier-Stokes equations

$D_u, D_v$ Diffusion coefficients for the activator and inhibitor in the Diffusion-Reaction equation, respectively

$\mathcal{F}$ Fourier transform of a signal