Training an FNO with TensorGRaD

This example compares three optimizers on Darcy-Flow:

AdamW baseline
TensorGRaD low-rank gradients with a 25% budget
TensorGRaD low-rank + sparse residual gradients with a matched 20% + 5% budget

The example follows the standard FNO Darcy workflow while using DarcyDataset(download=True) so it does not rely on repository-local data. It records train H1, validation H1, and validation L2 losses for all three methods and visualizes their predictions side by side.

# sphinx_gallery_thumbnail_number = 2

Import dependencies

from pathlib import Path
import sys

for path in [Path.cwd(), *Path.cwd().parents]:
    if (path / "neuralop" / "__init__.py").exists():
        repo_root = path
        break
else:
    repo_root = None

if repo_root is not None and str(repo_root) not in sys.path:
    sys.path.insert(0, str(repo_root))

import matplotlib.pyplot as plt
import numpy as np
import torch
from torch.utils.data import DataLoader

from neuralop import H1Loss, LpLoss, Trainer
from neuralop.data.datasets.darcy import DarcyDataset
from neuralop.models import FNO
from neuralop.training import (
    AdamW as BaselineAdamW,
    TensorGRaD,
    fno_tensorgrad_param_groups,
)
from neuralop.utils import count_model_params

plt.rcParams["figure.dpi"] = 110

Configuration

TensorGRaD is applied to the spectral convolution weights. Other FNO parameters, such as lifting/projection layers and skip connections, remain in a regular AdamW parameter group.

N_TRAIN = 1000
N_TEST = 100
BATCH_SIZE = 128
N_EPOCHS = 15
LEARNING_RATE = 1e-2
WEIGHT_DECAY = 1e-4
HIDDEN_CHANNELS = 24
N_MODES = (8, 8)
UPDATE_PROJ_GAP = 4
LOW_RANK_SCALE = 10.0
SPARSE_SCALE = 1.0
LAMBDA_SPARSE = 1.0
SPARSE_TYPE = "randk"

LOW_RANK_BUDGET = 0.25
LOW_RANK_SPARSE_LOW_RANK_BUDGET = 0.20
LOW_RANK_SPARSE_SPARSE_BUDGET = 0.05
assert abs(
    LOW_RANK_BUDGET
    - (LOW_RANK_SPARSE_LOW_RANK_BUDGET + LOW_RANK_SPARSE_SPARSE_BUDGET)
) < 1e-12

DATA_ROOT = Path.home() / ".cache" / "neuraloperator" / "darcy"
TEST_RESOLUTIONS = [16, 32]
TEST_BATCH_SIZES = [BATCH_SIZE, BATCH_SIZE]

torch.manual_seed(0)
np.random.seed(0)
device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
print(f"Using device: {device}")

Using device: cpu

Loading the Darcy-Flow dataset

DarcyDataset(download=True) downloads the public Darcy files from Zenodo if they are not already present in DATA_ROOT.

darcy_dataset = DarcyDataset(
    root_dir=DATA_ROOT,
    n_train=N_TRAIN,
    n_tests=[N_TEST, N_TEST],
    batch_size=BATCH_SIZE,
    test_batch_sizes=TEST_BATCH_SIZES,
    train_resolution=16,
    test_resolutions=TEST_RESOLUTIONS,
    encode_input=False,
    encode_output=True,
    download=True,
)

train_loader = DataLoader(
    darcy_dataset.train_db,
    batch_size=BATCH_SIZE,
    shuffle=True,
    num_workers=0,
    pin_memory=torch.cuda.is_available(),
)
test_loaders = {
    resolution: DataLoader(
        darcy_dataset.test_dbs[resolution],
        batch_size=batch_size,
        shuffle=False,
        num_workers=0,
        pin_memory=torch.cuda.is_available(),
    )
    for resolution, batch_size in zip(TEST_RESOLUTIONS, TEST_BATCH_SIZES)
}
data_processor = darcy_dataset.data_processor.to(device)
print(f"Loaded Darcy-Flow test resolutions: {list(test_loaders)}")

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Loading test db for resolution 16 with 100 samples
Loading test db for resolution 32 with 100 samples
Loaded Darcy-Flow test resolutions: [16, 32]

Creating the FNO model

def make_model():
    torch.manual_seed(0)
    return FNO(
        n_modes=N_MODES,
        in_channels=1,
        out_channels=1,
        hidden_channels=HIDDEN_CHANNELS,
        projection_channel_ratio=2,
    ).to(device)


preview_model = make_model()
print(f"FNO parameters: {count_model_params(preview_model):,}")
del preview_model

FNO parameters: 191,881

Creating the optimizers

The compressed-gradient budgets are matched: low-rank uses a 25% budget, and low-rank+sparse uses a 20% low-rank budget plus a 5% sparse residual budget.

def adamw_optimizer(model):
    return BaselineAdamW(
        model.parameters(),
        lr=LEARNING_RATE,
        weight_decay=WEIGHT_DECAY,
    )


def tensorgrad_low_rank_optimizer(model):
    param_groups = fno_tensorgrad_param_groups(
        model,
        rank=LOW_RANK_BUDGET,
        min_params=1000,
        update_proj_gap=UPDATE_PROJ_GAP,
        scale=LOW_RANK_SCALE,
    )
    return TensorGRaD(param_groups, lr=LEARNING_RATE, weight_decay=WEIGHT_DECAY)


def tensorgrad_low_rank_sparse_optimizer(model):
    param_groups = fno_tensorgrad_param_groups(
        model,
        rank=LOW_RANK_SPARSE_LOW_RANK_BUDGET,
        sparse_ratio=LOW_RANK_SPARSE_SPARSE_BUDGET,
        min_params=1000,
        update_proj_gap=UPDATE_PROJ_GAP,
        scale=LOW_RANK_SCALE,
        sparse_scale=SPARSE_SCALE,
        sparse_type=SPARSE_TYPE,
        lambda_sparse=LAMBDA_SPARSE,
    )
    return TensorGRaD(param_groups, lr=LEARNING_RATE, weight_decay=WEIGHT_DECAY)


print(f"Low-rank budget: {LOW_RANK_BUDGET:.0%}")
print(
    "Low-rank + sparse budget: "
    f"{LOW_RANK_SPARSE_LOW_RANK_BUDGET:.0%} + "
    f"{LOW_RANK_SPARSE_SPARSE_BUDGET:.0%} = "
    f"{LOW_RANK_BUDGET:.0%}"
)

Low-rank budget: 25%
Low-rank + sparse budget: 20% + 5% = 25%

Setting up losses

l2loss = LpLoss(d=2, p=2)
h1loss = H1Loss(d=2)

train_loss = h1loss
eval_losses = {"h1": h1loss, "l2": l2loss}

Training all three models

We subclass Trainer only to store epoch-wise train and validation metrics so we can plot all three methods on the same axes.

def to_float(value):
    if torch.is_tensor(value):
        return float(value.detach().cpu())
    if value is None:
        return None
    return float(value)


class HistoryTrainer(Trainer):
    def train_one_epoch(self, epoch, train_loader, training_loss):
        train_err, avg_loss, avg_lasso_loss, epoch_train_time = super().train_one_epoch(
            epoch,
            train_loader,
            training_loss,
        )
        self._latest_history_row = {
            "epoch": epoch,
            "step": (epoch + 1) * len(train_loader),
            "train_err": to_float(train_err),
            "avg_loss": to_float(avg_loss),
            "epoch_train_time": to_float(epoch_train_time),
        }
        return train_err, avg_loss, avg_lasso_loss, epoch_train_time

    def evaluate_all(
        self,
        epoch,
        eval_losses,
        test_loaders,
        eval_modes,
        max_autoregressive_steps=None,
    ):
        eval_metrics = super().evaluate_all(
            epoch=epoch,
            eval_losses=eval_losses,
            test_loaders=test_loaders,
            eval_modes=eval_modes,
            max_autoregressive_steps=max_autoregressive_steps,
        )
        row = dict(getattr(self, "_latest_history_row", {"epoch": epoch}))
        row.update({key: to_float(value) for key, value in eval_metrics.items()})
        self.history.append(row)
        return eval_metrics


def train_with_optimizer(name, optimizer_builder):
    torch.manual_seed(0)
    np.random.seed(0)
    model = make_model()
    optimizer = optimizer_builder(model)
    scheduler = torch.optim.lr_scheduler.CosineAnnealingLR(optimizer, T_max=N_EPOCHS)

    print(f"\n### {name} ###")
    print(f"Optimizer: {optimizer.__class__.__name__}")

    trainer = HistoryTrainer(
        model=model,
        n_epochs=N_EPOCHS,
        device=device,
        data_processor=data_processor,
        wandb_log=False,
        eval_interval=1,
        use_distributed=False,
        verbose=True,
    )
    trainer.history = []
    trainer.train(
        train_loader=train_loader,
        test_loaders=test_loaders,
        optimizer=optimizer,
        scheduler=scheduler,
        regularizer=False,
        training_loss=train_loss,
        eval_losses=eval_losses,
    )
    return model, trainer.history


optimizer_builders = {
    "AdamW": adamw_optimizer,
    "Low-rank 25%": tensorgrad_low_rank_optimizer,
    "Low-rank 20% + sparse 5%": tensorgrad_low_rank_sparse_optimizer,
}

models = {}
histories = {}
for name, optimizer_builder in optimizer_builders.items():
    models[name], histories[name] = train_with_optimizer(name, optimizer_builder)

### AdamW ###
Optimizer: AdamW
Training on 1000 samples
Testing on [50, 100] samples         on resolutions [16, 32].
/opt/hostedtoolcache/Python/3.13.16/x64/lib/python3.13/site-packages/torch/nn/modules/module.py:1794: UserWarning: FNO.forward() received unexpected keyword arguments: ['y']. These arguments will be ignored.
  return forward_call(*args, **kwargs)
Raw outputs of shape torch.Size([128, 1, 16, 16])
/home/runner/work/neuraloperator/neuraloperator/neuralop/training/trainer.py:536: UserWarning: H1Loss.__call__() received unexpected keyword arguments: ['x']. These arguments will be ignored.
  loss += training_loss(out, **sample)
[0] time=0.82, avg_loss=0.9276, train_err=115.9536
/home/runner/work/neuraloperator/neuraloperator/neuralop/training/trainer.py:581: UserWarning: LpLoss.__call__() received unexpected keyword arguments: ['x']. These arguments will be ignored.
  val_loss = loss(out, **sample)
Eval: 16_h1=0.7442, 16_l2=0.5149, 32_h1=0.7953, 32_l2=0.5110
[1] time=0.78, avg_loss=0.6149, train_err=76.8675
Eval: 16_h1=0.5251, 16_l2=0.3364, 32_h1=0.6150, 32_l2=0.3339
[2] time=0.81, avg_loss=0.4923, train_err=61.5434
Eval: 16_h1=0.4585, 16_l2=0.2929, 32_h1=0.5320, 32_l2=0.2855
[3] time=0.80, avg_loss=0.4188, train_err=52.3457
Eval: 16_h1=0.3952, 16_l2=0.2617, 32_h1=0.4870, 32_l2=0.2545
[4] time=0.80, avg_loss=0.3610, train_err=45.1285
Eval: 16_h1=0.3492, 16_l2=0.2305, 32_h1=0.4668, 32_l2=0.2353
[5] time=0.80, avg_loss=0.3204, train_err=40.0474
Eval: 16_h1=0.3173, 16_l2=0.2115, 32_h1=0.4216, 32_l2=0.2108
[6] time=0.82, avg_loss=0.2984, train_err=37.2995
Eval: 16_h1=0.2966, 16_l2=0.2019, 32_h1=0.4191, 32_l2=0.2160
[7] time=0.80, avg_loss=0.2731, train_err=34.1346
Eval: 16_h1=0.2752, 16_l2=0.1855, 32_h1=0.3987, 32_l2=0.1995
[8] time=0.80, avg_loss=0.2581, train_err=32.2678
Eval: 16_h1=0.2634, 16_l2=0.1770, 32_h1=0.3937, 32_l2=0.1921
[9] time=0.78, avg_loss=0.2498, train_err=31.2236
Eval: 16_h1=0.2570, 16_l2=0.1722, 32_h1=0.3792, 32_l2=0.1875
[10] time=0.79, avg_loss=0.2433, train_err=30.4120
Eval: 16_h1=0.2503, 16_l2=0.1666, 32_h1=0.3723, 32_l2=0.1807
[11] time=0.79, avg_loss=0.2389, train_err=29.8586
Eval: 16_h1=0.2449, 16_l2=0.1628, 32_h1=0.3713, 32_l2=0.1778
[12] time=0.80, avg_loss=0.2350, train_err=29.3775
Eval: 16_h1=0.2445, 16_l2=0.1626, 32_h1=0.3713, 32_l2=0.1786
[13] time=0.80, avg_loss=0.2336, train_err=29.1964
Eval: 16_h1=0.2423, 16_l2=0.1606, 32_h1=0.3696, 32_l2=0.1762
[14] time=0.87, avg_loss=0.2323, train_err=29.0437
Eval: 16_h1=0.2420, 16_l2=0.1603, 32_h1=0.3700, 32_l2=0.1762

### Low-rank 25% ###
Optimizer: TensorGRaD
Training on 1000 samples
Testing on [50, 100] samples         on resolutions [16, 32].
Raw outputs of shape torch.Size([128, 1, 16, 16])
[0] time=0.92, avg_loss=0.8374, train_err=104.6737
Eval: 16_h1=0.5911, 16_l2=0.3839, 32_h1=0.6585, 32_l2=0.3720
[1] time=0.90, avg_loss=0.5380, train_err=67.2530
Eval: 16_h1=0.4862, 16_l2=0.3014, 32_h1=0.5611, 32_l2=0.2885
[2] time=0.90, avg_loss=0.4543, train_err=56.7880
Eval: 16_h1=0.4172, 16_l2=0.2563, 32_h1=0.5052, 32_l2=0.2537
[3] time=0.90, avg_loss=0.3912, train_err=48.8942
Eval: 16_h1=0.3652, 16_l2=0.2335, 32_h1=0.4722, 32_l2=0.2343
[4] time=0.88, avg_loss=0.3499, train_err=43.7427
Eval: 16_h1=0.3592, 16_l2=0.2293, 32_h1=0.4595, 32_l2=0.2342
[5] time=0.90, avg_loss=0.3331, train_err=41.6413
Eval: 16_h1=0.3184, 16_l2=0.2123, 32_h1=0.4471, 32_l2=0.2214
[6] time=0.89, avg_loss=0.3054, train_err=38.1697
Eval: 16_h1=0.2977, 16_l2=0.1993, 32_h1=0.4172, 32_l2=0.2081
[7] time=0.91, avg_loss=0.2878, train_err=35.9769
Eval: 16_h1=0.2828, 16_l2=0.1915, 32_h1=0.4051, 32_l2=0.1973
[8] time=0.91, avg_loss=0.2843, train_err=35.5322
Eval: 16_h1=0.2791, 16_l2=0.1886, 32_h1=0.4032, 32_l2=0.1981
[9] time=0.87, avg_loss=0.2705, train_err=33.8180
Eval: 16_h1=0.2652, 16_l2=0.1796, 32_h1=0.3925, 32_l2=0.1890
[10] time=0.88, avg_loss=0.2619, train_err=32.7347
Eval: 16_h1=0.2577, 16_l2=0.1749, 32_h1=0.3925, 32_l2=0.1870
[11] time=0.90, avg_loss=0.2565, train_err=32.0658
Eval: 16_h1=0.2557, 16_l2=0.1737, 32_h1=0.3883, 32_l2=0.1847
[12] time=0.89, avg_loss=0.2535, train_err=31.6849
Eval: 16_h1=0.2534, 16_l2=0.1721, 32_h1=0.3875, 32_l2=0.1841
[13] time=0.86, avg_loss=0.2520, train_err=31.4969
Eval: 16_h1=0.2522, 16_l2=0.1714, 32_h1=0.3896, 32_l2=0.1846
[14] time=0.90, avg_loss=0.2511, train_err=31.3897
Eval: 16_h1=0.2519, 16_l2=0.1713, 32_h1=0.3883, 32_l2=0.1837

### Low-rank 20% + sparse 5% ###
Optimizer: TensorGRaD
Training on 1000 samples
Testing on [50, 100] samples         on resolutions [16, 32].
Raw outputs of shape torch.Size([128, 1, 16, 16])
[0] time=0.93, avg_loss=0.8865, train_err=110.8161
Eval: 16_h1=0.6933, 16_l2=0.4067, 32_h1=0.7337, 32_l2=0.4036
[1] time=0.91, avg_loss=0.5961, train_err=74.5099
Eval: 16_h1=0.5248, 16_l2=0.3204, 32_h1=0.5962, 32_l2=0.3026
[2] time=0.89, avg_loss=0.4829, train_err=60.3650
Eval: 16_h1=0.4465, 16_l2=0.2743, 32_h1=0.5290, 32_l2=0.2671
[3] time=0.90, avg_loss=0.4151, train_err=51.8914
Eval: 16_h1=0.3840, 16_l2=0.2454, 32_h1=0.4897, 32_l2=0.2443
[4] time=0.90, avg_loss=0.3994, train_err=49.9283
Eval: 16_h1=0.3611, 16_l2=0.2296, 32_h1=0.4656, 32_l2=0.2321
[5] time=0.89, avg_loss=0.3509, train_err=43.8585
Eval: 16_h1=0.3390, 16_l2=0.2189, 32_h1=0.4590, 32_l2=0.2234
[6] time=0.88, avg_loss=0.3258, train_err=40.7239
Eval: 16_h1=0.3115, 16_l2=0.2019, 32_h1=0.4351, 32_l2=0.2096
[7] time=0.88, avg_loss=0.3030, train_err=37.8715
Eval: 16_h1=0.2960, 16_l2=0.1928, 32_h1=0.4223, 32_l2=0.2012
[8] time=0.90, avg_loss=0.2865, train_err=35.8082
Eval: 16_h1=0.2844, 16_l2=0.1868, 32_h1=0.4151, 32_l2=0.1964
[9] time=0.88, avg_loss=0.2751, train_err=34.3890
Eval: 16_h1=0.2758, 16_l2=0.1813, 32_h1=0.4064, 32_l2=0.1906
[10] time=0.88, avg_loss=0.2681, train_err=33.5174
Eval: 16_h1=0.2672, 16_l2=0.1747, 32_h1=0.4075, 32_l2=0.1862
[11] time=0.87, avg_loss=0.2619, train_err=32.7406
Eval: 16_h1=0.2633, 16_l2=0.1718, 32_h1=0.4025, 32_l2=0.1839
[12] time=0.89, avg_loss=0.2579, train_err=32.2355
Eval: 16_h1=0.2600, 16_l2=0.1702, 32_h1=0.4013, 32_l2=0.1837
[13] time=0.90, avg_loss=0.2552, train_err=31.8969
Eval: 16_h1=0.2589, 16_l2=0.1694, 32_h1=0.4011, 32_l2=0.1832
[14] time=0.89, avg_loss=0.2542, train_err=31.7743
Eval: 16_h1=0.2587, 16_l2=0.1693, 32_h1=0.4007, 32_l2=0.1834

Reporting best validation losses

plot_resolution = max(test_loaders)
eval_h1_key = f"{plot_resolution}_h1"
eval_l2_key = f"{plot_resolution}_l2"


def best_metric(history, key):
    best_row = min(history, key=lambda row: row[key])
    return best_row[key], best_row["step"]


summary_rows = []
for name, history in histories.items():
    best_h1, best_h1_step = best_metric(history, eval_h1_key)
    best_l2, best_l2_step = best_metric(history, eval_l2_key)
    summary_rows.append(
        {
            "method": name,
            "best_val_h1": best_h1,
            "best_val_h1_step": best_h1_step,
            "best_val_l2": best_l2,
            "best_val_l2_step": best_l2_step,
        }
    )

header = (
    f"{'method':<28} {'best_h1':>12} {'h1_step':>8} "
    f"{'best_l2':>12} {'l2_step':>8}"
)
summary_lines = [header, "-" * len(header)]
for row in summary_rows:
    summary_lines.append(
        f"{row['method']:<28} "
        f"{row['best_val_h1']:>12.5g} "
        f"{row['best_val_h1_step']:>8} "
        f"{row['best_val_l2']:>12.5g} "
        f"{row['best_val_l2_step']:>8}"
    )

summary_text = "\n".join(summary_lines)
print(summary_text)

method                            best_h1  h1_step      best_l2  l2_step
------------------------------------------------------------------------
AdamW                             0.36957      112      0.17617      112
Low-rank 25%                      0.38753      104      0.18365      120
Low-rank 20% + sparse 5%          0.40073      120      0.18322      112

Loss curves

All panels use optimizer steps on the x-axis. Validation H1 and L2 are shown at the highest test resolution.

fig, axes = plt.subplots(1, 3, figsize=(15, 4))
for name, history in histories.items():
    steps = [row["step"] for row in history]
    axes[0].plot(steps, [row["avg_loss"] for row in history], marker="o", label=name)
    axes[1].plot(steps, [row[eval_h1_key] for row in history], marker="o", label=name)
    axes[2].plot(steps, [row[eval_l2_key] for row in history], marker="o", label=name)

axes[0].set_title("Train H1 loss")
axes[1].set_title(f"Validation H1 loss ({plot_resolution} x {plot_resolution})")
axes[2].set_title(f"Validation L2 loss ({plot_resolution} x {plot_resolution})")

for ax in axes:
    ax.set_xlabel("Optimizer step")
    ax.set_ylabel("Loss")
    ax.grid(True, alpha=0.3)
    ax.legend()

plt.tight_layout()
fig.show()

Train H1 loss, Validation H1 loss (32 x 32), Validation L2 loss (32 x 32)

Visualizing predictions

The columns compare the input, ground truth, AdamW prediction, low-rank prediction, and low-rank+sparse prediction.

test_samples = test_loaders[plot_resolution].dataset
prediction_labels = ["AdamW", "Low-rank 25%", "Low-rank 20% + sparse 5%"]

fig = plt.figure(figsize=(12, 7))
for index in range(3):
    data = test_samples[index]
    data = data_processor.preprocess(data, batched=False)

    x = data["x"].to(device)
    y = data["y"].to(device)

    with torch.no_grad():
        predictions = {
            label: models[label](x.unsqueeze(0)).squeeze().detach().cpu()
            for label in prediction_labels
        }

    panels = [
        ("Input x", x[0].detach().cpu(), "gray"),
        ("Ground truth y", y.squeeze().detach().cpu(), None),
        *[(label, predictions[label], None) for label in prediction_labels],
    ]
    for column, (title, image, cmap) in enumerate(panels):
        ax = fig.add_subplot(3, 5, index * 5 + column + 1)
        ax.imshow(image, cmap=cmap)
        if index == 0:
            ax.set_title(title)
        ax.set_xticks([])
        ax.set_yticks([])

fig.suptitle(
    f"Darcy-Flow FNO predictions at {plot_resolution} x {plot_resolution}",
    y=0.98,
)
plt.tight_layout()
fig.show()

Darcy-Flow FNO predictions at 32 x 32, Input x, Ground truth y, AdamW, Low-rank 25%, Low-rank 20% + sparse 5%

Total running time of the script: (2 minutes 40.569 seconds)

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