Title: Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning
Venue: NeurIPS 2024
Accepted: yes
Keywords: scientific machine learning

unsupervised pretraining

neural operators

foundation models
Abstract: Recent years have witnessed the promise of coupling machine learning methods and physical domain-specific insights for solving scientific problems based on partial differential equations (PDEs).  However, being data-intensive, these methods still require a large amount of PDE data. This reintroduces the need for expensive numerical PDE solutions, partially undermining the original goal of avoiding these expensive simulations. In this work, seeking data efficiency, we design unsupervised pretraining for PDE operator learning. To reduce the need for training data with heavy simulation costs, we mine unlabeled PDE data without simulated solutions,
and we pretrain neural operators with physics-inspired reconstruction-based proxy tasks. To improve out-of-distribution performance, we further assist neural operators in flexibly leveraging a similarity-based method that learns in-context examples, without incurring extra training costs or designs. Extensive empirical evaluations on a diverse set of PDEs demonstrate that our method is highly data-efficient, more generalizable, and even outperforms conventional vision-pretrained models. We provide our code at https://github.com/delta-lab-ai/data_efficient_nopt.

Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning

Abstract

Recent years have witnessed the promise of coupling machine learning methods and physical domain-specific insights for solving scientific problems based on partial differential equations (PDEs). However, being data-intensive, these methods still require a large amount of PDE data. This reintroduces the need for expensive numerical PDE solutions, partially undermining the original goal of avoiding these expensive simulations. In this work, seeking data efficiency, we design unsupervised pretraining for PDE operator learning. To reduce the need for training data with heavy simulation costs, we mine unlabeled PDE data without simulated solutions, and we pretrain neural operators with physics-inspired reconstruction-based proxy tasks. To improve out-of-distribution performance, we further assist neural operators in flexibly leveraging a similarity-based method that learns in-context examples, without incurring extra training costs or designs. Extensive empirical evaluations on a diverse set of PDEs demonstrate that our method is highly data-efficient, more generalizable, and even outperforms conventional vision-pretrained models. We provide our code at https://github.com/delta-lab-ai/data_efficient_nopt .

1 Introduction

Recent advancements in machine learning methodology have shown promise in solving partial differential equations (PDEs) [24, 67, 38, 45, 44, 50, 39, 25]. A significant development in this area is the concept of operator learning for PDEs. This approach differs from traditional neural network methods, which are restricted to fixed-dimension input and output, since neural operators focus on learning mappings between function spaces [38, 45, 44]. Like other neural network methods, neural operators are recognized to be universal approximators for any continuous operator [46, 38], enabling them to approximate any physical operator, including solution operators for various parametric PDE families. A solution operator is defined as a function that maps physical inputs to output solutions. Previous work has shown that in simple settings, neural operators can effectively capture complex, multi-scale dynamic processes [45, 46, 74, 73].

However, neural operators tend to suffer from a problem common to other deep networks, namely the need for enormous quantities of data . Limited availability of data is common in science and

38th Conference on Neural Information Processing Systems (NeurIPS 2024).

engineering. High-fidelity numerical simulations are computationally costly or even infeasible for many applications [71]. For example, an extreme-scale simulation of magnitude 7.0 earthquake at frequencies up to 10 Hz in San Francisco requires 3600 Summit GPU nodes and 42.7 hour [53].

Motivated by this data-efficiency challenge, recent works, particularly in natural language processing (NLP) and computer vision (CV), have focused on unsupervised (or self-supervised) pretraining 2 to reduce the cost of collecting or generating labeled data. Such pretrained models have been shown to be highly data-efficient in downstream fine-tuning [7, 29, 8], and they can even become few-shot learners without any downstream data [3]. In CV, researchers collect large amounts of natural images without any manual labels, and then pretrain visual encoders with proxy tasks, such as Noise-Contrastive Estimation (NCE) [59], masked reconstruction [28], rotation and jigsaw prediction [58, 20]. In NLP, people typically pretrain models via next-word prediction or masked tokens [3, 12].

However, unsupervised pretraining is still largely underexplored in Scientific Machine Learning (SciML). Therefore, our core question is: How can we design unsupervised pretraining for operator learning to reduce the data simulation costs?

In this work, we resort to unsupervised pretraining for neural operator learning to achieve data efficiency in SciML. We overview our framework in Figure 1. First, we define unlabeled data for PDEs, which avoids heavy computation costs for simulating PDE solutions. We propose two physics-inspired reconstruction-based proxy tasks, and we pretrain neural operators on unlabeled PDE data. We demonstrate that with unsupervised pretraining, our neural operators not only improve counterparts trained with more simulated data, but also they outperform off-the-shelf pretrained checkpoints from other popular domains (such as CV) that are ready for fine-tuning. Then, to further improve the data efficiency during out-ofdistribution (OOD) inference, we design a similarity-based method that learns in-context examples [3, 49, 80, 81, 47]. This approach introduces zero overhead during training: one just maintains the standard training pipeline, and it can be seamlessly plugged in for OOD inference, without further fine-tuning. In more detail, we summarize our main contributions:

Out-of-Distribution Inference

no training cost

no simulated solution

Fine-tuning

reduced simulation costs

···

Inputs

(

Unlabeled PDE Data

)

Pretrain with

Physics-Inspired

Proxy Tasks

In-Context

Learning

Demos

Train with

Inputs

Predicted Inputs

Predicted Solutions

Predicted Solution

Query

Inputs

Stage 1

Stage 2

Stage 3

We introduce unlabeled PDE data and unsupervised pretraining for data-efficient neural operator learning. We show that our method can achieve better performance than models trained with more simulated PDE solutions, or fine-tuned from public checkpoints pretrained on other benchmarks, demonstrating the importance of unsupervised pretraining on domain-specific PDE data.

We propose a similarity-based method to improve the OOD generalization of neural operators, which is flexible and can scale up to a large number of unseen in-context examples ('demos').

We provide detailed empirical evaluations on both diverse PDE benchmarks and also several realworld scenarios, demonstrating that we can achieve both strong forward modeling performance and significant savings in PDE simulations.

2 Related Works

2.1 Machine Learning for Scientific Modeling

There has been a long history of using learning-based methods to model physical scientific phenomena [41, 42, 6, 5]. A representative line of work is so-called physics-informed neural networks (PINNs) [67, 87, 19, 18, 69], which try to incorporate physics in neural networks by including the differential form of the PDE as an additional physics loss regularization term. However, this paradigm is confined to specific PDE scenarios (e.g., fixed PDE coefficients), instead of being more physics-agnostic. Moreover, recent work has highlighted several fundamental 'issues' with PINNbased methods [40, 15]. On the other hand, operator learning methods, including Fourier Neural Operators [45, 44, 38] and Deep Operator Network [50], have achieved progress in approximating the solution operators of PDEs. Although these data-driven approaches show promise in learning PDE solutions, they (like many neural network-based methods) rely on vast quantities of high-fidelity labeled data. For operator learning, such data are usually computationally expensive to simulate [67, 2, 84]. More recently, people have tried to generate synthetic PDE solutions to train SciML models [27]. In contrast, our method works on unlabeled PDE data. This approach is distinct from the generation of synthetic PDE data, and it could be further combined as a semi-supervised learning strategy.

2.2 Unsupervised Pretraining and Foundation Models

Unsupervised (or self-supervised) pretraining is a key method in CV and NLP to achieve meaningful representations [7], data-efficient fine-tuning [31], and foundation models [1]. In CV, contrastive learning learns meaningful features by distinguishing between similar (positive) and different (negative) samples [59, 76, 7, 56, 29]. Masked Autoencoder (MAE) [28] uses a reconstructive approach where parts of the input are masked and the model learns to predict masked parts. In NLP, among the most prominent works are large language models (LLMs) such as GPT [3, 64, 65] and BERT [12, 66], which leverage token predictions for pretraining. Similar directions also show progress in SciML. For example, [2] and [54] propose to create augmented views in the solution space via Lie Symmetries; [73] study the scaling behavior of supervised pretraining and OOD generalization, charting directions for foundational models for SciML; [43] target learning astronomical foundation models with crossmodal contrastive learning; and [52] build large task-agnostic models with a broad understanding of common physical behavior to serve as foundation models for SciML.

2.3 In-Context Learning (ICL)

In-context learning (ICL) is a promising paradigm that helps deep networks generalize to unseen domains with a few in-context examples. Early works in CV seek to learn feature-level correspondence between the target and a few 'shots,' such that models can generalize to open-set unseen objects [17, 83]. In NLP, people find LLMs are naturally few-shot learners [3], and thus tuning or optimizing prompts becomes extremely important to improve the in-context learning performance of LLMs [85, 72]. More recently, within SciML, a different operator learning strategy, termed 'in-context operator learning,' has been proposed [80, 81, 47]. During both training and inference, the neural operator is asked to make predictions by explicitly leveraging a predefined number of so-called 'demo' examples (pairs of physical parameters and simulated solutions). This approach provides a balance between model generalization and addressing data scarcity in the scenario of OOD testing.

3 Methods

In this section, we introduce our framework (outlined in Figure 1). We propose first to pretrain the model with unsupervised pretraining (Sec. 3.1), which will contribute to the data efficiency and reduced PDE simulation costs during standard training of neural operators. When we move to OOD scenarios during inference, we test our models with in-context examples (Sec. 3.2) to avoid further fine-tuning costs.

3.1 Unsupervised Pretraining

The core idea in unsupervised (or self-supervised) pretraining is to train a neural network with properly designed proxy tasks. These proxy tasks do not require labeled data, but they are designed

to be highly related to the supervised learning objectives of interest. While popular in CV and NLP, unsupervised pretraining on unlabeled data has never been explored in PDE operator learning in SciML. This is due to two unresolved questions : 1) What kinds of unlabeled data can we use to train neural operators? 2) How to design proxy tasks for PDE data? We address these questions below.

3.1.1 Unlabeled PDE Data

How to Define Unlabeled PDE Data? In general, when a neural operator is trained on PDE datasets [45, 74, 73], it learns to map inputs (physical parameters, coordinates, forcing functions, initial conditions, etc.) to PDE solutions. Therefore, given a set of PDE data (collected via simulations or observations), its unlabeled version is defined as the one without PDE solutions. Our unlabeled PDE data is a broader concept of related inputs in modeling PDE systems. Let us consider the second-order linear differential equation as a general example. It is formulated as

where x ∈ R n represents physical space that varies in different systems (e.g. n = 3 for 2D timedependent PDEs); the coefficients (or physical parameters) a ij , b i , c are known from the physical process; u is the target solution; and f denotes an external forcing function [57]. We can consider two situations where solutions are unavailable:

Time-independent equations: following [45, 73], our unlabeled PDE data include physical parameters ( a ij , b i , c ), forcing functions ( f ), and coordinates (grids of the discrete physical space).

Time-dependent equations (e.g., forecasting problems [74, 52]): without simulating the temporal dynamics, our unlabeled PDE data only include initial snapshot u 0 ( x ) that defines PDE systems. Note that collecting snapshots with temporal dynamics in large-scale scenes is more complex than capturing individual snapshots. For example, weather forecasting [30] and smoke dispersion [14] require continuous monitoring and multiple sensors, whereas single measurements are simpler and less resource-intensive. Long-term data collection often involves extensive networks and processing, unlike one-time measurements.

For concrete examples of different PDEs and unlabeled data that we will study, see Appendix A. There are two main reasons for pretraining only on unlabeled PDE data, as discussed below.

Cheap Generation of Unlabeled PDE Data. One critical reason that leads to expensive computational costs when collecting PDE data is the time marching scheme [23, 22] in numerical simulation. However, only generating unlabeled PDE data and snapshots without temporal dynamics will be much cheaper than simulating solutions (Table 4), making our unsupervised pretraining highly feasible in practice. Our pretraining strategy will be very data-efficient, and can avoid the heavy computational cost of simulating complex time-dependent equations for massive high-fidelity labeled solutions.

Benefits of Pretraining on Unlabeled PDE Data. Beyond the cheap generation, pretraining on unlabeled PDE data has the following benefits. First, regularization against overfitting . Unsupervised pretraining can strongly regularize the model towards better generalization. Second, faster convergence . Pretraining on unlabeled PDE data will provide neural operators with domain-adapted initializations and can accelerate training speed. Third, meaningful representations . Pretraining on unlabeled PDE data can help models extract useful representations for subsequent operator learning. We defer our results and experimental details in Figure 4 in Sec. 4.1.

3.1.2 Proxy Tasks

To illustrate our general approach of constructing proxy tasks, we choose two variants of reconstruction as our core proxy tasks. In particular, we will input unlabeled PDE data to our neural operators, and after a decoder network, we will force the output to be close to the input. We consider two perturbation variants (or augmented views). They are inspired by real-world settings when people collect scientific data, and they are important invariances we need to introduce to SciML models.

Masked Autoencoder. Masked autoencoders (MAEs) have been shown to be scalable selfsupervised learners [28]. The method is conceptually simple: remove a portion of the input data, and

learn to predict the removed content. These methods enable training in NLP and CV of generalizable models containing over one hundred billion parameters [12, 3]. Here, we investigate the potential of MAE for scientific modeling. Unsupervised Pretraining

Motivation: PDE dynamics are invariant to sparse sensing of the full field scientific data. It is very common [4, 16, 32] that scientific data need to be collected from sparse sensors, and that people need to reconstruct or generate data for domains without sensors. We enforce our model to learn sensor invariance via random masking, and we extract the invariant features over distorted views of the same unlabeled PDE data. The invariance to sparse sensing of scientific data will facilitate the robustness of the representations from MAE.

Mask

Reconstruction

Blur

Super-resolution

Augmented

Inputs

Inputs

(Unlabeled PDE Data)

Predicted Inputs

Therefore, we consider MAE as a proxy task. Specifically, our MAE is a straightforward autoencoding approach that reconstructs the original signal, given its partial observation. Like all autoencoders, our approach has an encoder that maps the observed signal to a latent representation, and a decoder that reconstructs the original signal from the latent representation. We randomly sample masks according to a certain masking ratio (i.e., the ratio of removed areas), and the values of masked areas of unlabeled PDE data are set to zero. Our loss function computes the mean squared error (MSE) between the reconstructed and original input. We compute the loss only on masked areas; if no mask is applied (i.e., a vanilla autoencoder), the MSE loss will be applied to all spatial areas.

Super-resolution. Super-resolution (SR) techniques have emerged as powerful tools for enhancing data resolution, improving the overall quality and fidelity of data representation, and retrieving fine-scale structures. SR is a task that involves recovering fine-scale data from corresponding coarsegrained data. It is also a popular task on PDE learning [9, 68], which is to train SciML models to preserve the inherent physical properties of scientific data.

Motivation: Numerical solutions of PDEs are expected to exhibit invariance to filtering blur or different resolutions of inputs [37, 79]. For instance, in turbulence simulations, the traditional numerical methods always fail to model the expected physical phenomenon with low-resolution meshes due to substantial numerical errors. SR has emerged as a powerful tool for subgrid modeling of PDE dynamics, especially helping to capture the critical patterns of turbulence [51, 37]. Given a specific input distribution, after fitting the SR objective, neural operators are expected to preserve the inherent physical properties and exhibit invariance to filtering blur.

Therefore, we introduce SR as another proxy task. Our objective shares the same motivation as recent SciML works for SR [68]. Specifically, we enforce the model to learn invariant features of unlabeled PDE data that are immune to resolution and blur. Blurry snapshots often occur when the resolution is too low to accurately represent the details in the original content. To do so, we apply a Gaussian filter to blur the unlabeled PDE data, and the autoencoder will reconstruct the high-resolution input with fine-grained details. Instead of applying a fixed blurring, we randomly sample the variance of the Gaussian filter from a certain range as augmentations.

3.1.3 PDEs

After pretraining on unlabeled PDE data, we fine-tune neural operators on simulated solutions of PDEs. We study two time-independent PDEs (Poisson, Helmholtz) and two time-dependent PDEs (Reaction-Diffusion, Navier-Stokes). We include details of these PDEs in Appendix A.

3.1.4 Model Architectures

We consider two popular architectures for fair comparisons with previous works. These are encoderdecoder architectures designed to reconstruct the original input given partial observations, where the encoder maps observed unlabeled PDE data to a latent space, and the decoder reconstructs the original conditions. We include visualizations of these architectures in Appendix D.

Fourier Neural Operator. Fourier Neural Operator (FNO) targets learning PDE data in the Fourier space. The original model backbone (encoder) employs Fourier transform and learns lower Fourier modes with linear transforms. The FNO backbone outputs features back to the spatial domain (i.e., the embeddings are on the pixel level). We refer readers to the original paper for details [45, 46].

Pretraining : We build the decoder to be identical to the encoder (except for the input/output dimension). Unlabeled PDE data are randomly masked at the pixel level.

Fine-tuning : After the pretraining, we discard the decoder, and we follow the original design to append two fully-connected layers (with ReLU activations) to predict final spatial-wise solutions.

Transformer. Transformers, which mainly employ self-attention and linear transform blocks, have shown promise in both NLP and CV [78, 13]. Different from FNO, which directly operates on grids, transformers tokenize and group grids into patches, i.e., each tokenized patch embeds a local neighborhood of subgrids. We follow the 3D transformer architecture of Video-MAE [77, 52]. Our encoder embeds patches by a linear projection with added positional embeddings (just as in a standard ViT), and it then processes the resulting set via a series of Transformer blocks. For the transformer, the unlabeled PDE data are randomly masked at the patch level.

Pretraining : For the transformer encoder, we only apply it on the subset of tokens that are visible (i.e., unmasked patches), and masked patches are removed. This allows us to train encoders efficiently. The input to the MAE decoder is the full set of tokens consisting of (i) encoded visible patches, and (ii) the mask token. The mask token is a shared and learned vector that indicates the presence of a missing patch to be predicted. We add positional embeddings to all tokens in this full set. Without this, mask tokens would have no information about their location in the input. Following [28, 77], we adopt an asymmetric design where the decoder is more lightweight (shallower and narrower) than the encoder.

Fine-tuning : After the pretraining, the decoder is preserved during fine-tuning, since we need to reconstruct the tokenized patches back to the input.

3.2 Similarity-based Mining of In-Context Examples

Out-of-distribution (OOD) generalization is a critical technical challenge, not only in SciML but also across multiple domains in AI for science [84]. To improve the OOD generalizability of neural operators and to reduce the extra effort of downstream fine-tuning, the following inference paradigm has been proposed: given a query input, the model is also provided with a few supporting examples (dubbed 'demos'), together with their ground-truth solutions, to make the final prediction. This approach enables the 'open-set' generalization of the model to make predictions on unseen samples.

Originally, in the literature on few-shot learning [82, 83, 55, 70, 35, 48, 61], people developed delicate architectures to find a correspondence between the input and the supporting examples. The purpose of this extra architecture/training design is twofold: first, find similarities between the target input and supporting examples; and second, aggregate labels of supporting examples for the final predictions. Recent ICL works [80, 81, 47] on learning PDE data also adopt this strategy, with transformers and cross-attention layers.

However, the ICL (in-context learning) of LLMs (large language models) enables a different strategy. The pretraining is still standard and simple (next/masked token prediction), without additional training costs. During inference, LLMs can auto-regressively take any number of few-shot examples, finding similarities between tokens in few-shot examples and those in the target query (via self-attention), and then generate responses by aggregating embeddings of tokens in few-shot examples. This ICL strategy used in LLM is highly scalable and training-efficient.

Motivated by this, we propose to leverage in-context examples via two steps (Algorithm 1).

Similarity by Prediction. We find spatial-wise and temporal-wise similar demos by calculating their distance in the output space. That means, for two input locations over the spatial and temporal domains, if we find their outputs of the trained neural operator similar, then we treat them as similar samples. Following [80, 81], we assume demos share the same distribution of physical parameters with the query.

Aggregation. For each spatial-temporal location of the query, after finding its similar samples in demos, we aggregate and average their solutions as the prediction.

Algorithm 1: Pseudocode of Similarity-based Mining of In-Context Examples.

1 Data resolution: x-axis ( W ), y-axis ( H ), output temporal steps ( T ), output channel dimensions for the solution ( C out ).

2 Input: Query input ( x ). Paired unlabeled PDE data ( X ) and solutions ( Y ∈ R J × H × W × T × C out ) as J demos. Trained Neural Operator Model M . TopK ( k ) demo solutions to aggregate.

3 ˆ y = M ( x )

4 ˆ Y = M ( X )

5 ˆ δ = ˆ y .reshape ( -1 , 1 , C out ) -ˆ Y .reshape (1 , -1 , C out ) H × W × T × ( J · H · W · T ) × C out

6 ˆ δ = absolute ( ˆ δ ) . sum ( -1)

7 index = argsort ( ˆ δ, -1)[: , : , : , : k ] ▷ Spatial-wise and temporal-wise selection of demos similar to the query.

8 ˆ y icl = take_along_dim( Y .reshape ( -1 , C out ) , index) ▷ Spatial-wise and temporal-wise aggregation of solutions from similar demos. Shape: H × W × T × C out × k .

9 Return: ˆ y icl .mean ( -1)

4 Empirical Results

To illustrate the benefits of our approach, we perform empirical evaluations on both PDE benchmarks and real-world observations. Most importantly , our unsupervised pretraining (on unlabeled PDE data, followed by fine-tuning) outperforms neural operators trained from scratch, while requiring fewer PDE simulations (Sec. 4.1 and Sec. 4.2). Moreover , our in-context examples can help the model generalize better to OOD cases (Sec. 4.3). In our experiments, we trained models three times with different random seeds for statistical significance. For more experimental details, see Appendix B. We also include ablation studies about pretraining hyperparameters in Appendix J. For visualizations of our pretraining, see Appendix K.

(a)

(b)

(c)

(d)

(e)

(f)

~2x10^3

savings

~2x10^3

savings

~6x10^3

savings

~5x10^2

savings

~8x10^4 savings

Poisson (FNO)

Helmholtz (FNO)

Reaction-Diffusion

Error

-十-unsupervised

+-

random init.

1.00

060

0.80

0.70

(FNO,PDE-Bench)

-+-unsupervised

-+- random init.

0.10

0.60

Relative

0.10

0.50

0.40

0.30

0.01

0.01

0.20

十

unsupervised

random init.

0.15

102

103

104

102

103

104

103

104

Number of TrainingSamples

Number of Training Samples

2D Incompressible Navier-Stokes

2D Incompressible Navier-Stokes

2D incompressible Navier-Stokes

(FNO-3D,PINODataSet)

(Video-MAE,PDE-Bench)

(Video-MAE,PDE-Bench)

0.50

0.40

unsupervised

0.20

--unsupervised

0.07

Error

0.30

random init.

random init.

vision pretrained (SSv2)

0.06

8881

Relative

2

0.20

0.04

0.05

0.07

0.06

0.05

0.04

十

random init.

unsupervised

vision pretrained (SSv2)

0.08

0.03

0.03 -

R

0.08

0.07

0.02

0.02

0.06

0.05

101

Number of Training Samples

102

103

102

Number of Training Samples

103

104

105

1

2

Rollout

3

4

5

▷ Shape: H × W × T × C out

▷ Shape: J × H × W × T × C out

▷ Query-Demo Distance. Shape:

4.1 Unsupervised Pretraining Enables Data-Efficient Operator Learning

Data Efficiency. We first demonstrate that, by leveraging unsupervised pretraining, neural operators can achieve improved errors with less simulated data. In Figure 3, on each PDE, compared with directly training from scratch ('random init.'), pretraining on unlabeled PDE data can help neural operators achieve better performance, which can further help reduce the amount of simulated data. Specifically, in our experiments, when we target achieving the best test error of the baseline ('random init.'), our method can save 5 × 10 2 ∼ 8 × 10 5 simulated solutions across diverse PDE systems.

Among all PDEs, we find that Helmholtz (Figure 3 (b)) is the most challenging. Both two curves failed to improve the error until we increased the number of simulated data points to over 1024. Meanwhile, the generalization gaps remain high (Figure 12 (b)), indicating low training errors. We suspect that learning on the Helmholtz equation may be exhibiting the grokking issue [63, 86], where the network quickly memorizes the training data, but the improvement in generalizability is delayed.

Unsupervised Pretraining Outperforms Off-the-Shelf Pretrained Checkpoints. Pretraining on unlabeled PDE data is not the only way to save simulation costs. As pretraining is widely adopted in CV, pretrained checkpoints on vision data become publicly available and are ready for fine-tuning. We choose to compare with Video-MAE [77] for state-of-the-art video understanding pretrained on SSV2 [21]. As shown in Figure 3 (e), vision-pretrained Video-MAE can only outperform the random initialization with high volumes of simulated data, while its performance suffers when fine-tuned with limited simulations. In contrast, our unsupervised pretraining on unlabeled PDE data can save a significant amount of simulated data.

As errors during testing may quickly accumulate with further timestamps, we also report results with more unrolled steps. We use checkpoints trained with the largest amount of simulated data from above for this study. As shown in Figure 3 (f), at each rollout step, our unsupervised pretraining achieves much better performance. Similarly, vision-pretrained Video-MAE is eventually outperformed by the random initialization at the long rollout step.

Benefits of Pretraining on Unlabeled PDE Data. Pretraining on unlabeled PDE data is beneficial beyond achieving better performance with reduced simulations. First , when training on extremely low volumes of data, neural operators tend to overfit, resulting in poor generalization. In Figure 4-left, pretraining on unlabeled PDE data can reduce the generalization gap (testing error -training error). We further show that better generalization gaps persist across all PDEs we studied (see Figure 12). Second , pretraining on unlabeled PDE data can lead to faster convergence during fine-tuning. Unlike standard random initializations from Gaussian distributions, pretraining on unlabeled PDE data will provide neural operators with domain-adapted initializations and facilitate a much faster convergence rate, as shown in Figure 4-middle 3 . Third , unsupervised pretraining can also help models extract useful representations for subsequent operator learning. From Figure 4-right, we find that even with pre-extracted features (i.e., fixed encoder and only fine-tuned decoder), our neural operators can still outperform the baselines where both the encoder and decoder are updated during fine-tuning.

2DIncompressibleNavier-Stokes

2DIncompressibleNavier-Stokes

2DIncompressibleNavier-Stokes

Train)

(Video-MAE,PDE-Bench)

(Video-MAE,PDE-Bench)

(Video-MAE,PDE-Bench)

Error(Training)

-+-

unsupervised

100

unsupervised

0.20

-+-

unsupervised

一

0.100

random init.

vision pretrained (SSv2)

random init.

(Test

Error

vision pretrained (SSv2)

random init.

Error

vision pretrained (SSv2)

8:08

unsupervised (encoder fixed)

Generalization Gap

10-1

2

80'0

0.07

0.06

Relative

0.010

Relative

0.05

Relative L2

0.04

0.03

102

0.02

0.001

102

NumberofTrainingSamples

103

104

105

0

100

200

Epoch

300

400

500

102

NumberofTrainingSamples

103

104

105

4.2 More Comprehensive Experiments on Real-World Data

We now move to a broader range of benchmarks. We will study real-world and noisy data instead of toy datasets, providing even more comprehensive experiments. These benchmarks are widely studied in previous works [60, 62, 52].

Datasets. We brief the background, with more details in Appendix L and visualizations in Fig. 15.

ECMWF Reanalysis v5 (ERA5) [30] is a public extensive dataset, which delivers hourly data on multiple atmospheric variables spanning from 1979 to today. ERA5 represents a type of atmospheric reanalysis dataset [34], integrating observations from a variety of measurement sources with numerical models through data assimilation [33]. Essentially, it reconstructs the most accurate estimation of the Earth's atmospheric conditions over time. This dataset has been extensively utilized in prior SciML studies [60, 68]. We focus on forecasting the important and challenging temperature atmospheric variable.

ScalarFlow [14] is a reconstruction of real-world smoke plumes. It assembles the first large-scale dataset of realistic turbulent flows. The availability of a large, volumetric data set opens up a wide range of applications, including re-simulations, novel visualization, and metric evaluations. The dataset contains 104 real-world smoke flow density reconstructions. Each reconstruction is captured from five different viewpoints for 150 temporal frames spanning 2.5 seconds.

Airfoil [75] is a large-scale dataset that contains the pressure and velocity simulations of the flow around real airfoils. This dataset has 53880 samples for training and 90 samples for testing. Each sample contains 6 channels. The 3 input channels include the binary spatial mask of the airfoil and the initial velocity of the freestream condition in the x and y directions. The output channels include the pressure and the x and y velocity components from the simulation at the steady state.

Model and Training. For time-dependent ERA5 (Sec. L.1) and ScalarFlow (Sec. L.2) datasets, we adopt the same VideoMAE architecture [77] used in Sec. 4.1. We use 15 consecutive temporal snapshots to forecast the next time step. We train VideoMAE with Adam, with other hyperparameters the same as in Table 3 column 'N.S. (PDEBench)'. For the time-independent steady-state Airfoil dataset (Sec. L.3), we adopt the 2D-FNO architecture. We train 2D-FNO with Adam, with other hyperparameters the same as in Table 3 column 'Poisson'.

Results. As shown in Figure 5, compared with directly training operators from scratch ('random init.'), pretraining on unlabeled data (2D snapshots of ERA5/ScalarFlow without temporal dynamics, or freestream velocities of Airfoil) can help neural operators achieve better performance on both temperature/flow forecasting and predictions of the steady-state pressure and velocity around airfoils.

ERA5(Video-MAE)

ScalarFlow(Video-MAE)

Airfoil (FNO)

Error

0.22

-O-unsupervised

=O-random init.

0.30

-O-unsupervised

=O-random init.

0.10

-O-

-O-

random init.

unsupervised

1

0.08

Relative

0.16

0.10

0.06

0.04

8=0

0.12

0.03

0.02

101

102

103

101

102

103

104

102

103

104

Number of Training Samples

NumberofTrainingSamples

4.3 In-Context Examples Enable Data-Efficient OOD Generalization

We now move to OOD settings, where models will be tested on PDE data simulated with physical parameters unseen during fine-tuning/training. We include how to simulate OOD samples in

Appendix B.2. Neural operators suffer from poor OOD generalization [73, 80]. Traditionally, improvements heavily depend on further fine-tuning on simulated data, which requires extra simulation and training costs. We study the benefits of leveraging test-time in-context examples. As shown in Figure 6, when we flexibly scale up the number of demos, we can keep improving FNO's OOD generalization on diverse PDEs. We follow [80, 81] that demos are randomly sampled from the same distribution used to generate the OOD test set. When the number of demos is 0, we have the baseline in the OOD setting. Notably, we introduce zero training overhead: we keep the standard training pipeline, and our mining of in-context examples can be seamlessly plugged in during OOD inference.

We further provide a baseline, which uses features extracted by the backbone of the neural operator (high-dimensional features before the final output layer) to find similar samples. As we can see, this baseline is worse than our method (both performance and confidence), indicating that the final output of the neural operator can more accurately indicate true similar samples.

See Appendix M for further discussions about the benefits of leveraging in-context examples, and Appendix N for visualizations.

0

5

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30

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2

3

4

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6

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2 Error

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FNO Output)

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1.0

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1.4

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2 Error

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Backbone Feature)

Ours (Similarity by

FNO Output)

0.975

1.000

1.025

2D Incompressible Navier-Stokes

(FNO-3D, PINO Dataset)

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Backbone Feature)

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0.425

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2 Error

5 Conclusion

In this work, we focus on improving the data efficiency of solving partial differential equations (PDEs) using deep learning, with a particular emphasis on unsupervised pretraining and in-context learning (ICL) methods. Our key contributions include introducing unsupervised pretraining for operator learning and a flexible ICL approach that enhances out-of-distribution (OOD) generalization without increasing training costs. Through extensive evaluations, we demonstrate that our method is not only more data-efficient, but it also achieves greater generalizability compared to existing approaches. By improving the data efficiency of neural operators for solving PDEs, our approach can significantly reduce the computational costs and energy demands of high-fidelity numerical PDE simulations. Additionally, by making advanced PDE solutions more accessible through efficient pretraining, our method has the potential to accelerate scientific and engineering progress across various fields, ultimately benefiting society. We hope our work will inspire the scientific machine learning (SciML) community to further address the high simulation costs and limited OOD generalization of neural operators, contributing to advancements that support both scientific innovation and environmental sustainability.

6 Limitations

Current limitations of our work: 1) We could design more physics-inspired proxy tasks and data augmentation methods for scientific data; 2) We could study more PDE systems in our unsupervised pretraining and in-context learning; 3) We could consider more different neural operator architectures. We expect that addressing these limitations will lead to broader impacts in future works.

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A PDEs

We consider two time-independent PDEs (Poisson, Helmholtz) and two time-dependent PDEs (Reaction-Diffusion, Navier-Stokes), as described below. We also describe types of unlabeled data for per-PDE examples. For a general definition of unlabeled PDE data, please refer to Section 3.1.1.

Poisson : Weconsider a two-dimensional (2D) elliptic PDE that arises in various physical situations, with periodic boundary conditions within a spatial domain Ω = [0 , 1] 2 :

where u ( x ) represents the solution, f ( x ) acts as the source (e.g., forcing) function, and x denotes spatial coordinate. The diffusion coefficient tensor, denoted by K , is employed to measure the physical properties of this system.

Unlabeled PDE Data : In [73], inputs to neural operators have four channels: the source function, and three diffusion coefficients (two diagonal elements and one off-diagonal element in the symmetric diffusion coefficient tensor) expanded to the whole spatial domain. We use this input as the unlabeled data to pretrain the neural operator.

Helmholtz : We consider the 2D inhomogeneous Helmholtz equation, which is a time-independent form of the wave equation, with periodic boundary conditions and a spatial domain Ω = [0 , 1] 2 . The governing equation is given by

where u ( x ) denotes the solution, f ( x ) represents the source function, and ω > 0 is the wavenumber (that defines the dynamic properties of the Helmholtz equation). This system produces high-frequency large wavenumber oscillatory patterns, which poses challenges in terms of generalization.

Unlabeled PDE Data : In [73], inputs to neural operators have two channels: the source function, and the wavenumber ω expanded to the whole spatial domain. We use this input as the unlabeled data to pretrain the neural operator.

Reaction-Diffusion: The 2D Reaction-Diffusion (RD) equation involves the interaction between two nonlinear coupled variables, i.e., the activator u ( t, x, y ) and the inhibitor v ( t, x, y ) . The RD equation is given by

where { D u , D v } are diffusion coefficients for u and v , respectively, and where R u and R v are reaction functions, which are defined as the Fitzhugh-Nagumo equation [36]:

The spatial domain considered for the 2D RD equation is Ω = [ -1 , 1] 2 , and the time duration is t ∈ (0 , 5] .

Unlabeled PDE Data : In PDEBench [74], we forecast the activator u and the inhibitor v (i.e. two input channels) with T = 10 . Since each individual snapshot can serve as an initial condition for forecasting, during our unsupervised pretraining we discard the temporal dynamics and use randomly shuffled snapshots of u and v with T = 1 as unlabeled PDE data. That means, during pretraining the model only has access to single and individual snapshots without any awareness of temporal dynamics.

Navier-Stokes Equation: Lastly, we consider the 2D incompressible Navier-Stokes equation in vorticity form on the unit torus, which is formulated as

Here, the velocity v is defined within the time duration [0 , T ] and the spatial domain [0 , 1] 2 (the vorticity can be formulated as w = ∇ × v ). Moreover, ρ is the density, and the coefficient

ν signifies the viscosity of a fluid, and f is the external forcing function. Dirichlet boundary conditions are employed in this system.

Unlabeled PDE Data : We follow the training settings from previous works. 1) For FNO [45], we focus on mapping the vorticity from the initial condition to the full solution ( w 0 ↦→ w | [0 ,T ) , with T = 33 ). The input includes four channels: the vorticity (which is the initial condition, duplicated for T times), and three (xy-spatial and temporal) linear mesh position embedding channels. Thus, similar to Reaction-Diffusion above, during pretraining, the model will take individual snapshots of vorticity and position embeddings as unlabeled data. 2) For PDEBench [74], we forecast both xy velocities and pressure (i.e. three input channels), with T = 15 . The model will take individual snapshots of vorticity and xy velocities as unlabeled data.

We summarize detailed inputs and outputs in Table 1.

B Detailed Experiment Settings

B.1 Distributions of Unlabeled PDE Data

In Table 2, for the purpose of OOD testing, we summarize the distribution of our unlabeled PDE data during pretraining, fine-tuning, and inference with in-context examples. Ranges of these physical parameters are inspired by [73]. During pretraining, we consider a wide distribution of unlabeled PDE data. When training (fine-tuning) our model, we consider in-distribution unlabeled PDE data. Finally, we test our similarity-based method that learns in-context examples in OOD settings. 4 For Helmholtz OOD, we choose a narrow range of coefficients ([15, 20]) mainly because its solution is very sensitive to the wavenumber, and FNO's performance significantly drops when we move to more extreme OOD settings.

B.2 Data Generation

Unlabeled PDE Data. We generate data for Poisson and Helmholtz [73], Reaction-Diffusion on PDE-Bench [74] and 2D incompressible Navier-Stokes on PINO Dataset [46] following the procedure mentioned in the paper. For unlabeled data generation, we bypass the computation of solvers, which expedites the generation speed, as shown in Table 4.

OOD Samples. The OOD data generation procedure is similar to the unlabeled data, except for the changes in the physical parameters coefficients. For Poisson and Helmholtz, we consider changing the range of diffusion eigenvalue and waver number respectively. For Navier-Stokes equation, we change the Reynolds number. We list the coefficients in Table 2.

B.3 Training Hyperparameters

We summarize our hyperparameters used during pretraining and fine-tuning/training in Table 3. These hyperparameters strictly follow previous works [45, 73, 74, 46, 52]. We conducted our experiments on four A100 GPUs, each with 40GB of memory.

C Examples of Simulation Costs

In Table 4, we demonstrate the cheap simulation of only unlabeled PDE data, versus simulating both unlabeled PDE data and solutions, on 2D incompressible Navier-Stoke on PINO Dataset [46] and Reaction-Diffusion on PDE-Bench [74]. We can see that unlabeled PDE data are extremely cheap to simulate. Therefore, our pretraining method can boost the performance and meanwhile save the heavy cost of data simulations.

D Model Architectures

In Figure 7, We show visualizations of architectures described in Sec. 3.1.4. Specifically, the FNO has 67.1M parameters, and the Video-MAE has 23.4M. During pretraining, FNO costs 20 GPU hours, and Video-MAE costs 18 GPU hours. During fine-tuning, FNO costs 4 GPU hours, and Video-MAE costs 6 GPU hours.

E Comparison with Contrastive Learning

Contrastive learning is an important self-supervised pretraining technique studied in computer vision. For a fair comparison, we directly compare with MoCo v2 [8], a highly-cited self-supervised learning method also originally implemented in PyTorch, whose core method is closely related to SimCLR [7] (originally implemented in TensorFlow).

We compare MoCo v2 with our method on a broader real-world benchmark ERA5 [30]. As shown in Figure 8, our unsupervised learning method can largely outperform MoCo v2. This extra comprehen-

Input

Linear

FFT

Linear

IFFT

Linear

GeLU

5

✕

Linear

GeLU

Linear

Output

IFFT: Inverse Fourier Transform

FNO

+

Conv3D

Positional Embedding

Tokens

Encoder Transformer Block

(width=384, #heads = 6)

Linear

Decoder Transformer Block

(width=192, #heads = 3)

12

✕

4

✕

Output

Input

VideoMAE

sive result demonstrates that our method can be widely adopted in real-world problems, outperforming previous unsupervised learning methods.

10 1

10 2

10 3

Number of Training Samples

0.12

0.14

0.16

0.18

0.20

Relative

2 Error

ERA5 (Video-MAE)

unsupervised

MoCo v2

F More Comparisons with Vision Pretrained Models

Beyond Figure 3(e) for the transformer on time-dependent PDE (Navier-Stokes), we further verify that pretraining on unsupervised PDE data makes FNO outperform its off-the-shelf vision-pretrained checkpoints. Specifically, we first pretrain the 2D FNO model on ImageNet [11], then fine-tune on the downstream PDE simulation data. As shown in Figure 9, vision-pretrained FNO performs worse during downstream fine-tuning on time-independent PDEs (including both Poisson and Helmholtz equations), confirming that domain-specific pretraining, even on unsupervised PDE data, is more beneficial than conventional checkpoints pretrained on unrelated domains like computer vision.

G Joint Pretraining Further Improves Performance

We also study if joint pretraining on unlabeled multiple PDE data can bring extra benefits. We combine all 46,080 unlabeled Poisson samples and all 46,080 unlabeled Helmholtz samples (see Table 3). We choose this setting because recent works on SciML foundation models [52, 26] also use all samples from each PDE for pretraining. We do zero-paddings for mismatched channels. From Figure 10, we can see that joint pretraining can further improve the performance of fine-tuning on different PDEs.

10 2

10 3

10 4

Number of Training Samples

0.01

0.10

Relative

2 Error

Pretrain: ImageNet

Finetune: Poisson (FNO)

unsupervised

random init.

vision pretrained

10 2

10 3

10 4

Number of Training Samples

0.01

0.10

1.00

Relative

2 Error

Pretrain: ImageNet

Finetune: Helmholtz (FNO)

unsupervised

random init.

vision pretrained

10 2

10 3

10 4

Number of Training Samples

0.01

0.10

Relative

2 Error

Finetune: Poisson (FNO)

unsupervised

random init.

unsupervised joint

10 2

10 3

10 4

Number of Training Samples

0.01

0.10

1.00

Relative

2 Error

Finetune: Helmholtz (FNO)

unsupervised

random init.

unsupervised joint

10 1

10 2

Number of Training Samples

2×10 1

3×10 1

4×10 1

6×10 1

Relative

2 Error

Pretrain: NS + RD

Finetune: RD (FNO)

unsupervised

random init.

unsupervised joint

H Fine-tuning on Unseen PDEs is Challenging

We also try to fine-tune neural operators on unseen PDEs (i.e. PDEs different from pretraining). Mismatched channels are padded with zeros. We find this will lead to worse performance compared with models pretrained on the same PDE, as shown in Figure 11.

10 2

10 3

10 4

Number of Training Samples

0.01

0.10

1.00

Relative

2 Error

Pretrain: Poisson

Finetune: Helmholtz (FNO)

unsupervised

random init.

unsupervised poisson

I Consistently Improved Generalization

Beyond the generation gap we have shown in Figure 4 (left) for the Navier Stokes equation from PDEBench, we further collect generation gaps of our models learned on other PDEs. As shown in Figure 12, on diverse PDE systems, our method can contribute to universally reduced overfitting.

(a)

(b)

(c)

(d)

Generalization Gap (Test - Train)

PoisSon (FNO)

Helmholtz (FNO)

!

-十-

unsupervised

1.00

random init.

Relative l2 Error

0.10

0.10

0.01

0.01

十-

unsupervised

0.00

-十-

random init.

102

103

104

102

103

104

Number of Training Samples

Number of Training Samples

Train)

Reaction-Diffusion

2DIncompressibleNavier-Stokes

(FNO,PDE-Bench)

(FNO-3D,PINODataset)

卜

0.40

0.50

-十-unsupervised

Relative 2 Error

-+- random init.

0.30

-十- random init.

0.40

OE'O

0.20

0.20

1888

20:0

0.06

0.05

0.04

0.03

0.05

103

104

101

102

103

Number of Training Samples

J More Ablation Studies

J.1 Magnitude of Perturbations during Pretraining.

We study the optimal magnitude of perturbations during pretraining: 'Mask Ratio' (1 indicates no visible grids, and 0 means no masks); and 'Blur Sigma' for the variance of Gaussian kernel for blur (larger the more degradation). We show our results in Table 5, 6, 7, 8. For example, on 2D incompressible Navier-Stokes with FNO, as shown in Table 8, we can find that when training with a low volume of data, we should use much stronger perturbations (high masking ratios and strong blur), whereas a high volume of data only requires mild perturbations.

J.2 Ablation of the Number of Pretraining Samples.

As shown in Table 9, the more unlabeled PDE data we use for pretraining, the better quality the pretrained model will be.

source

mask

prediction

error

X10-3

0.6

0.6

4.4

Pois.

0.2

0.2

2.2

-0.2

-0.2

V0.0

X10-3

0.8

0.8

1.1

Helm.

0.4

0.4

0.6

-0.0

-0.0

V0.0

0.4

0.4

0.3

R

-0.0

-0.0

0.1

D

-0.4

-0.4

V0.0

20.1

20.1

33.4

s

-11.7

-11.7

16.7

-43.6

-43.6

V0.0

K Visualization of MAE Pretraining

To demonstrate the efficacy of our MAE-based pretraining, we show the unlabeled PDE data and its reconstructed version in Figure 13 (MAE pretraining on different PDEs) and Figure 14 (MAE pretraining on 2D incompressible Navier-Stokes with varying mask ratios). We can see that all inputs are accurately reconstructed with low errors and similar patterns.

0.1

source

mask

prediction

error

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-32.9

-9.2

14.6

-32.9

-9.2

14.6

0.0

16.2

32.5

-23.9

-3.6

16.8

-23.9

-3.6

16.8

0.0

11.9

23.8

-12.6

1.8

16.2

-12.6

1.8

16.2

0.0

10.6

21.2

-15.8

-0.3

15.2

-15.8

-0.3

15.2

0.0

12.0

24.0

-16.9

3.4

23.7

-16.9

3.4

23.7

0.0

20.0

40.0

-11.3

3.5

18.3

-11.3

3.5

18.3

0.0

9.0

18.0

-43.6

-11.7

20.1

-43.6

-11.7

20.1

0.0

16.7

33.4

-48.7

1.6

51.9

-48.7

1.6

51.9

0.0

40.3

80.6

-38.2

-9.3

19.6

-38.2

-9.3

19.6

0.0

28.0

56.1

8

res

L Details and Visualizations of Real-World Data

L.1 ECMWFReanalysis v5 (ERA5)

We utilize data from 2006 to 2015, with the snapshots taken every 6 hours and a spatial resolution of 360 × 360 . The total number of snapshots is 14600. We apply the mean-standard deviation normalization to the data and downsample the snapshots to a spatial resolution of 180 × 180 . We split 75% of the data for pretrain and 25% for finetune. For each split, we further separate 80% of the data for training, 10% for validation, and 10% for testing.

L.2 ScalarFlow

The original spatial resolution is 1062 × 600 . We crop the snapshots to 900 × 600 to remove the padding and background. We remove the first 15 timeframes for each simulation to avoid the initial transient phase at the beginning of the smoke flow generation. We then downscale the snapshots to 180 × 120 and apply mean-standard deviation normalization to the data. With a total of 70200 snapshots, we split 80% of the data for pretrain and 20% for finetune. For each split, we further separate 80% of the data for training, 10% for validation, and 10% for testing.

L.3 Airfoil

We split 80% of the training data for pretrain and 20% for finetune. The original spatial resolution of each sample is 128 × 128 . We crop the snapshots to 96 × 45 to remove the background. We then apply min-max normalization channel-wise to the sample.

We show visualizations of real-world scientific data in Figure 15.

Temperature (Kelvin)

Density

(e) Freestream X

(f) Freestream Y

(g) Mask

(h) Pressure

(i) Velocity X

(j) Velocity Y

0.0

15.1

30.3

0.0

6.1

12.1

0.0

0.5

1.0

-2124.1

-947.0

230.1

-39.5

8.8

57.1

-8.4

20.6

49.6

310

279

248

(a)t=0

(b) t =17

(c)t=50

(d)t=67

M Benefits of In-context Examples

We try to understand the benefit of leveraging in-context examples by decomposing the relative MSE error into 'scale' and 'shape.' 'Scale' is the slope of a linear regression between targets and predictions (closer to 1 the better), indicating the alignment of the range of model outputs with targets. 'Shape' is the normalized relative MSE (i.e., model outputs or targets are normalized by their own largest magnitude before MSE), indicating the alignment of scale-invariant spatial/temporal structures. We show results in Figure 16. We find that the benefit of in-context examples lies in that the scale of the model's output keeps being better calibrated ('scale' being closer to 1) when adding more demos.

In numerical simulations or predictions of PDEs, there are settings where the scale or magnitude of solutions is more important than the exact shape/pattern:

Heat Transfer: In large-scale systems, the focus might be on overall temperature and extreme values. For instance, in evaluating a cooling system, the key concern might be the peak temperature rather than the detailed temperature distribution.

Fluid Dynamics: For applications like aerodynamics, the overall drag or lift force on an object is often more critical than capturing every detail of the flow pattern, such as in airfoil design.

Environmental Modeling: The concentration of pollutants at specific locations or total pollutant transport is often more crucial than the exact distribution, such as in groundwater flow studies.

0

5

10

15

20

25

30

Number of Demos

0.25

0.30

0.35

0.40

0.45

Scale (Closer to 1 the Better)

Scale (Slope of Linear Regression)

Shape (Normalized Relative Error)

0.70

0.72

0.74

0.76

0.78

0.80

0.82

0.84

Shape (Smaller the Better)

FNO on Poisson

0

5

10

15

20

25

30

Number of Demos

0.2

0.0

0.2

0.4

0.6

Scale (Closer to 1 the Better)

Scale (Slope of Linear Regression)

Shape (Normalized Relative Error)

1.12

1.14

1.16

1.18

1.20

1.22

1.24

Shape (Smaller the Better)

FNO on Helmholtz

0

5

10

15

20

25

30

Number of Demos

1.00

1.05

1.10

1.15

1.20

1.25

Scale (Closer to 1 the Better)

Scale (Slope of Linear Regression)

Shape (Normalized Relative Error)

0.45

0.46

0.47

0.48

0.49

0.50

0.51

Shape (Smaller the Better)

FNO-3D on PINO Dataset

(2D incompressible Navier-Stokes)

N Visualizations with In-Context Examples

We show visualizations of our similarity-based mining of in-context examples in Figure 17. In this visualization, we find that the range of numerical solutions (e.g., values in colorbars) predicted with in-context examples becomes closer to the target. Meanwhile, based on this visualization, we conclude that OOD generalization is challenging for neural operators because of: 1) Significantly different patterns of solutions under different physical parameters; 2) Different value ranges of solutions under different physical parameters.

Pois.

target

#demo = 0

#demo = 1

#demo = 32

Helm.

N.S.

-0.1

-0.0

0.1

-0.2

0.2

0.5

-0.3

0.0

0.4

-0.1

0.0

0.2

-0.2

-0.0

0.1

-0.4

-0.0

0.4

-0.2

-0.0

0.1

-0.1

-0.0

0.1

-122.6

-26.0

70.5

-75.0

-21.2

32.6

-103.3

-26.3

50.7

-110.6

-30.9

48.9

?

O

C

8.

2

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the problem of training neural operators to solve forward partial differential equation (PDE) problems in data-limited settings. The paper motivates this setting with the large data simulation cost incurred by these methods when generating a large training set of PDE solutions. The paper suggests a self-supervised pretraining method derived from self-supervised learning tasks popular in the computer vision community. This pretraining method only requires a dataset of initial conditions, PDE coefficients, and physical parameters; it does not require any PDE solutions. The paper shows the benefit of their method by showing their pretrained models outperform randomly-initialized models or models pretrained on general computer vision tasks; there are many evaluations across a range of PDE problems and real-world experimental data. In addition, the paper considers an \"out-of-distribution\" (OOD) task where PDE coefficient settings are changed at test time and suggests an \"In-Context Learning\" (ICL) strategy, which uses the trained model to adaptively combine examples of ground-truth solutions.\n\nSTRENGTHS:\nThis paper addresses important challenges in the field of neural operators and provides well-motivated solutions to these problems. There are a large number of experiments to quantify the effect of these solutions. I will list individual strengths below:\n- The paper considers important problems in the field of neural operators: reducing the dependence on training data generated by PDE solvers, adapting trained models to out-of-distribution problem settings, and quantifying performance on complicated real-world datasets.\n- The experimentation in this paper is very thorough. The paper shows experiments on PDE problems across a broad range of physical parameters as well as real-world datasets ERA5 temperature, ScalarFlow, and Airfoil.\n- The paper's pretraining methods are relatively simple and can be applied to a broad class of problems. It is easy to imagine these methods being used by other members of the community.\n- The paper's proposed ICL method is a creative emulation of attention mechanisms that is applicable to general neural operator methods.\n\nWEAKNESSES:\nI have concerns about the experimental design and worry the experimental results are overstated in the text. The claims about reducing data simulation costs are particularly overstated, and I believe the paper could be improved by contracting its stated goals to focus on real-world problem settings where data is scarce, and simulation methods are unavailable. I will list individual weaknesses below:\n- The paper motivates the limited-data regime due to the computational burden of generating PDE solutions. The stated goal is to \"reduce data simulation costs\". However, the method suggests a pre-training procedure that trains full-sized models on $\\geq 40,000$ samples for $500$ epochs. This also introduces a large computational cost, and this added cost of pre-training is never addressed. Without accounting for this added cost of pre-training, the paper's claims about reducing data simulation costs feel vacuous.\n- The paper fails to contextualize their results on the PDE benchmarks with meaningful baselines. Figures 3 and 5 only consider random initialization of neural network weights and pretrained on a general computer vision task as baselines. This ignores the simple baseline of generating extra samples with the same computational cost as pretraining.\n- Similarly, the paper fails to contextualize the results of the OOD experiment with meaningful baselines. The description of the OOD experiment indicates that the methods in citations [76, 77] are designed to solve the same problem. When reading the text, it seems like these methods could be used as baselines for comparison in Figure 6. If this is not the case, because [76, 77] are solving a weaker version of the problem or require significantly more data or compute resources, this is an advantage of the proposed method and should be highlighted in the paper.\n- The amount and type of unlabeled pretraining data in the ERA5 and ScalarFlow experiments are unusual for a self-supervised learning setting, which raises questions about the significance of this experiment. I have written specific questions about this below.\n- The results of the OOD experiment indicate the OOD task is still a difficult problem for neural operators, but the text does not discuss this difficulty. Without such a discussion, the significance of this experiment and the ICL method is unclear.\n\nQUESTIONS:\nGeneral:\n - Line 132 states \"initial snapshot $u_0(x)$ ... is sufficient for defining PDE systems.\" Could the authors explain or qualify this statement? The definition of a PDE system must also include a partial differential equation with parameters, a domain, and possibly boundary conditions like $u_0(x)$.\n - Line 185 states \"Numerical solutions of PDEs are expected to exhibit invariance to filtering blur or different resolutions of inputs.\" Could the authors explain or qualify this statement? There are many examples where qualitative features of PDE solutions depend on the discretization level. One example is the Helmholtz problem considered with high wavenumber and forcing term with high-frequency oscillations. As the resolution of the input forcing term and solution increase, higher-frequency oscillations in the solution can be resolved. For the Poisson equation, one can show analytically that applying a blur to the forcing term will change the solution.\n - The experiment in Appendix E shows encouraging results that the general-purpose self-supervised learning method MoCo v2 does not perform well on the ERA5 task. I believe that if one were to combine Figure 8 and Figure 5a, the result would show that pretraining with MoCo v2 produces almost no improvement over random initialization. This seems like a nice result and wonder if the authors have intuition about why this is the case.\n\nQuestions about PDE experiments:\n - What are the inputs and outputs of the model for each experiment? As I was reading this paper, I was constantly asking this question and spent time jumping around in the appendix looking for details. Would it be possible to collect all of this information in a table for ease of reference?\n - For each PDE experiment, what is the distribution of forcing functions (Poisson, Helmholtz) or initial conditions (Reaction-Diffusion, Navier-Stokes)?\n - How does the \"diffusion\" physics parameter in Table 1 relate to the definition of the Poisson equation in Equation 1? Are elements of K randomly drawn from this range?\n - How does the Reynolds number relate to the definition of the Navier-Stokes equations (Equation 5)?\n - Are there spatial boundary conditions for the Reaction-Diffusion equation?\n - What do the error bars in Figures 3 and 4 represent? Are they showing the min/max of the three independent random seeds?\n\nQuestions about real-world data experiments:\n - The ERA5 and ScalarFlow datasets have temporal observations; the learning task is to predict the state of the system at time $t+16$ given a sequence of state observations at times $[t, t+15]$. What is the unlabeled data used in pretraining for these cases? Is it sequences of state observations at times $[t, t+15]$, for a set of different values $t$?\n - For these experiments, how does using snapshots at $t > 0$ agree with the definition of unlabeled PDE data for time-dependent systems in Section 3.1.1?\n - These experiments use a large number of snapshots to construct the pretraining dataset. In the ERA5 example, this is enough data to build $14,600 \\times \\frac{75}{100} \\times \\frac{1}{16} \\approx 680$ labeled samples. In the ScalarFlow example, this is enough data to build $70,200 \\times \\frac{80}{100} \\times \\frac{1}{16} \\approx 3500$ labeled samples. Could the snapshots used in the pretraining dataset be used in a supervised learning framework? If so, why doesn't the “random init” baseline have access to this data?\n - Are there examples of simulation or experimental settings where it is reasonable to assume we can easily gather snapshots of a system state, but these snapshots could not be used in a supervised neural operator framework? Mentioning such examples in the text would help readers understand the significance of this experiment.\n\nQuestions about the ICL method and OOD experiments:\n - Do the authors have any intuition for why the Poisson and Helmholtz OOD performance is poor, and the Navier-Stokes performance is better? Providing information about the types of PDE problems amenable to in-context learning would be a valuable contribution and would increase the impact of this experiment.\n - Is there a connection between the expected smoothness of a particular PDE’s solution and the performance of the ICL method on that PDE? For example, we know that the solution of the Poisson equation should be relatively smooth, but the proposed ICL method outputs highly non-smooth estimates.\n - I see that ICL with more demos improves the scale of the prediction. Are there problem settings where the scale of the solution is of particular interest? Discussing such settings may better contextualize the results on these challenging OOD tasks.\n\n\nAs they continue their work, I want to ensure the authors are aware of this preprint, which provides a fast method for generating synthetic data without a PDE solver, applicable to certain PDE problems. Fitting the neural operator to synthetic data could be a useful pretraining task if the PDE is known. I am NOT asking the authors to compare their method with this preprint:\n\nE. Hasani and R. A. Ward, “Generating synthetic data for neural operators.” arXiv, Jan. 04, 2024. Available: http://arxiv.org/abs/2401.02398\n\nLIMITATIONS:\nThe authors have addressed some limitations of their work. As mentioned above, I believe the computational expense of the pretraining stage is a limitation of this work which is not discussed in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-02"}

comments: SUMMARY:
This paper proposes a pre training strategy for operator learning. They introduce unsupervised training in the case of physical data. 2 architectures are studied: transformers and FNO. Finally, some experiments on different PDEs are conducted to highlight the properties of the model.

STRENGTHS:
The paper is well-written and explains step-by-step the algorithms and architecture. 
Several experiments highlight the performances of the model on both simulated and real-world datasets, which is a challenging setting. Moreover, statistics are provided for each experiment.

WEAKNESSES:
As mentioned in the paper, masked auto-encoding has already been proposed in CV applications. The contribution however is unclear for me. ICL and masked-auto-encoding are widely known techniques. Moreover, it is difficult to identify why is the unsupervised pretraining performs best than other baselines.

QUESTIONS:
-	For the unlabeled data for time-dependent PDE, how does the PDE ICs are sufficient to identify the PDE? How does the model have access to the PDE parameters? 
-	Is your model trained on multiple physics? Or pretraining are proceeded on one PDE? 
-	On figure3, why are the unsupervised training performing better than training from scratch? How is proceeded this “random init” training? Is it using data with supervised loss? 
-	What is the comparison of other baselines in the OOD setting? 
-	What is the computational cost of training your model compared to other baselines? How many parameters do the baselines contains?

LIMITATIONS:
Author have proposed a discussion on the limitations of their paper. However, they did not address societal impact and ecological considerations.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-11

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper aims at improving the data efficiency of deep learning models for tackling Operator Learning. The paper focuses on two aspects: 1) They pretrain neural operators on data that do not assume labels, i.e. without the target function or the trajectory solution of states. To do so, they rely on a masked auto-encoding task or a super-resolution task, that both can be trained without labels. They show that this pretraining strategy can effectively improve the sample efficiency compared to a random initialization.  2) They propose an algorithm to do in-context learning at test time based on a number of demo examples and show results on out-of-domain PDEs.\n\nSTRENGTHS:\n* The main strength of the paper is that the pretraining strategy effectively reduces the number of samples to reach a certain target accuracy compared to a random initialization.\n* The method is tested against multiple equations from different levels of difficulty.\n* The comparison with pretrained vision transformers, though a little counterintuitive, is informative on the capacity of transferability of this kind of models to different tasks.\n* The ICL experiments seem to be consistent, the more demos the better the results.\n\nWEAKNESSES:\n* Overall the paper is quite difficult to read, there are many different aspects that are described within the method section and it can be hard to understand in the experiment section which is what.\n* If I am not mistaken there is no consistent comparison between the different tasks and architectures of the method (super-resolution vs masking x FNO vs transformer) for the different datasets. \n* The ICL method should be compared to at least one baseline. I suggest to do a simple kernel method but on the inputs rather than with the predictions of FNO. It can be a kNN or anything, but it would actually give a good understanding of how well the method works.\n\nQUESTIONS:\n* What is the best effective strategy for pretaining ? Is it the masking-damasking or super-resolution that yields the best results ? Or do you use both ?\n* Did you observe a difference between the video transformer and FNO for this setup ? Is there one that you would recommend better than the other ?\n\nLIMITATIONS:\nThere is a limitation section.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents an unsupervised pretraining approach for PDE solvers based on Meta Autoencoding and Super Resolution. The authors show that after pretraining, the model can achieve better accuracy than training a solver from scratch. This paper also presents an \"in-context learning\" strategy for inference on out-of-distribution data.\n\nSTRENGTHS:\n- Designing pretraining techniques for SciML foundation model is an important research direction.\n\n- To the best of my knowledge, the proposed pretraining technique is new in the SciML context.\n\nWEAKNESSES:\n- **Some important experimental setups are missing**. In the main body of the paper, it's not even clear what the model size is, what the pretraining datasets are, and whether the model is pretrained on one class of PDEs or multiple classes of PDEs.\n  - A follow-up concern is that according to Table 2 in Appendix B, it seems the authors pretrain different models for different PDEs. This significantly undermines the value of pretraining, because the ultimate goal of pretraining is to obtain a general purpose foundation model which can be easily transferred to different tasks (i.e., different PDEs in this context). If the method requires pretraining for each single class of PDEs, it's not efficient any more, and the comparison in Fig. 3 is not fair because **the proposed approach is still a problem-specific technique** and the pretrained models require additional training.\n\n- **Missing baselines.** For time-independent PDEs, the MAE for image should also be included as a baseline to support the claim of _outperforming conventional vision-pretrained models_.\n\n- I feel that **\"in-context learning\" in Sec. 3.2 is a misnomer**. First, there is no notion of \"context\" in the pretrained model. Second, the presented model is not an auto-regressive model. Third, unlike the self-attentive models, the presented model does not actively or flexibly \"learn\" from the examples; the way to aggregate information is fixed given Algorithm 1. The presented \"ICL\" algorithm deviates too much from the standard practice. I strongly encourage the authors to get rid of  \"ICL\" and refers the proposed approach as a few-shot learning strategy in the paper to avoid misleading the community.\n\n- In Sec. 4.3, the few-shot learning accuracy looks poor, and the standard deviation of the error is not reported.\n\n- Some discussions in the papers are inaccurate or lack justification.\n  - Line 95: BERT does not leverage next token prediction for pretraining.\n  - Line 142: Multiple benefits of pretraining are listed, but no references are provided, and no relevant experiments are presented to justify those arguments (e.g., better generalization, faster training speed, etc).\n  - Line 158: the authors cite the BERT paper and claim that \"_these methods enable training in NLP and CV of generalizable models containing over one hundred billion parameters_\". I don't think any variant of BERT is scaled to one hundred billion parameters.\n\n- Other issues.\n  - Line 26: DNN $\\to$ deep neural network\n  - Line 32: has  $\\to$  have\n  - Line 46: start   $\\to$  start with\n\nQUESTIONS:\n- Can you report the model sizes, pretraining cost, and finetuning cost?\n\n- Can the pretrained model be applied to other PDEs which are not seen in pretraining?\n\nLIMITATIONS:\nThe authors discuss some of the limitations of this paper in Sec. 6.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-13"}