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

Authors: wuyang  chen
jialin  song
pu  ren
shashank  subramanian
dmitriy  morozov
michael w mahoney
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 18:10:56

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.

Main paper: 
 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?Fine-tuning reduced 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:Inputs1. 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. 2. 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"). 3. 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 
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.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.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. 
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.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.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 asn i,j=1 a ij (x)u xixj + n i=1 b i (x)u xi + c(x)u = f (x),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. We defer our results and experimental details in Figure 4 in Sec. 4.1.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. 70% ) of spatial locations are masked, followed by a Gaussian blur. After the encoder and decoder, the full set of input is required to be reconstructed.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.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.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.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.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] 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. Data resolution: x-axis (W ), y-axis (H), output temporal steps (T ), output channel dimensions for the solution (Cout). 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. ▷ Spatial-wise and temporal-wise aggregation of solutions from similar demos. Shape:ŷ = M(x) ▷ Shape: H × W × T × Cout Ŷ = M(X) ▷ Shape: J × H × W × T × Cout δ = ŷ.reshape(-1, 1, Cout) -Ŷ .reshape(1, -1, Cout) ▷ Query-Demo Distance. Shape: H × W × T × (J • H • W • T ) × Cout δ = absolute( δ).H × W × T × Cout × k. Return: ŷicl .mean(-1) 
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. 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 errortraining 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. 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. 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.Figure 5 : For real-world scientific problems, pretraining neural operators on unlabeled PDE data improves its performance and data efficiency. We study VideoMAE [77] pretrained with unlabeled snapshots (no temporal dynamics), and then fine-tune across different numbers of temporal snapshots on ERA5 (left) and ScalarFlow (middle). We also pretrain 2D-FNO [45] on freestream velocities and fine-tune on time-independent steady-state airflow pressure and velocities (right). "random init.": models are trained from scratch with random initialization.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.  
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. 
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.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 . 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.  
broader impacts: 
 Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [NA] .Justification: This work is about foundational research on data augmentations. It does not lead to any societal impact because: 1) our motivations, goals, and results are too distant from social applications to make any social impact; 2) we mainly use public data and benchmarks; and 3) we use public model architectures and follow previous works which have open access.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.• Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA] .Justification: The paper poses no such risks because all datasets studied in our work are from public benchmarks, and we have clearly cited original papers.Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] .Justification: All datasets studied in our work are from public benchmarks, and we have clearly cited original papers.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.• If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. • If this information is not available online, the authors are encouraged to reach out to the asset's creators. • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: 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 .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. 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. 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- sive result demonstrates that our method can be widely adopted in real-world problems, outperforming previous unsupervised learning methods. 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.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. 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 . 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. 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. As shown in Table 9 , the more unlabeled PDE data we use for pretraining, the better quality the pretrained model will be. 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. 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.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.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 . 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:1. 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. 2. 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. 3. 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. Justification: Our work does not involve any potential harm, societal impact, or violation.Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).In this paper, we use the terms "unsupervised pretraining" and "self-supervised pretraining" interchangeably.Training curves when the number of training samples of 14760.For Navier Stokes from PDEBench, we use the original data. We could not generate our own pretraining/finetuning data with different ranges of physics parameters, due to a possible mismatch of the provided configuration files and the version of Phiflow used in PDEBench (see GitHub issue at https://github.com/pdebench/PDEBench/issues/36). 


Decision: The authors consider the problem of training neural operators to solve forward partial differential equation (PDE) problems in data-limited settings, motivated by the large data simulation cost incurred by these methods when generating a large training set of PDE solutions. They propose a self-supervised pretraining method derived from tasks popular in the computer vision community, and demonstrate the benefit of their method on a variety of PDE problems and real-world experimental data, compared to other self-supervised baselines. Since the majority of operator learning approaches require “supervised” data obtained by solving the forward PDE, most baselines they compare to are to not necessarily state-of-the-art (only SotA for the unsupervised setting, which has been less studied). The authors also show experiments on an "out-of-distribution" (OOD) task where PDE coefficient settings are changed at test time and an in-context learning strategy is used together with their pretrained models to improve performance relative to baselines, as in in-context operator learning.

After the rebuttal and discussion, reviewers’ scores for the paper increased, converging towards accept. The authors provided numerous clarifications on unclear details, especially methodological and experimental, in the paper during the rebuttal, as well as new experiments which convinced reviewers of the solidity of the paper’s methodology. Reviewers expressed some concerns about the competitiveness of the authors’ experimental baselines, but acknowledge that they are reasonable in the setting of self-supervised pretraining for operator learning that the authors consider, which has been relatively less explored than the supervised case. The AC concurs with the reviewers judgment on these points, seeing the paper as providing a nontrivial contribution to a potentially important research direction (especially in light of the clarifications of computational savings given in the response to u3i6’s review), and therefore recommends acceptance. The authors should not fail to incorporate all improvements in presentation and new experimental results recommended by the reviewers and presented during the rebuttal into the revision in a prominent way (i.e., not just in the appendices), especially regarding the proposed ICL method (per comments of oEEz and oUb1, the clear dependence on the in-context operator learning approach should be emphasized in Section 3.2), results on computational savings given to u3i6, and all clarifications of protocol made in the discussion, especially with u3i6.

Reviews: review: paper_summary: paper_summary: This 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.
strength_weakness: strength_weakness: - **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.
  - 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.

- **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_.

- 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.

- In Sec. 4.3, the few-shot learning accuracy looks poor, and the standard deviation of the error is not reported.

- Some discussions in the papers are inaccurate or lack justification.
  - Line 95: BERT does not leverage next token prediction for pretraining.
  - 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).
  - 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.

- Other issues.
  - Line 26: DNN $\to$ deep neural network
  - Line 32: has  $\to$  have
  - Line 46: start   $\to$  start with
questions: questions: - Can you report the model sizes, pretraining cost, and finetuning cost?

- Can the pretrained model be applied to other PDEs which are not seen in pretraining?
limitations: limitations: The authors discuss some of the limitations of this paper in Sec. 6.
paper_summary: paper_summary: This 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.
strength_weakness: strength_weakness: * 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.
* 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. 
* 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.
questions: questions: * 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 ?
* 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 ?
limitations: limitations: There is a limitation section.
paper_summary: paper_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.
strength_weakness: strength_weakness: 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: 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: limitations: Author have proposed a discussion on the limitations of their paper. However, they did not address societal impact and ecological considerations.
paper_summary: paper_summary: This 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.
strength_weakness: strength_weakness: I 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:
- 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.
- 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.
- 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.
- 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.
- 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.
questions: questions: General:
 - 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 $u_0(x)$.
 - 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.
 - 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.

Questions about PDE experiments:
 - 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?
 - For each PDE experiment, what is the distribution of forcing functions (Poisson, Helmholtz) or initial conditions (Reaction-Diffusion, Navier-Stokes)?
 - 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?
 - How does the Reynolds number relate to the definition of the Navier-Stokes equations (Equation 5)?
 - Are there spatial boundary conditions for the Reaction-Diffusion equation?
 - What do the error bars in Figures 3 and 4 represent? Are they showing the min/max of the three independent random seeds?

Questions about real-world data experiments:
 - 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 $t$?
 - 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?
 - 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?
 - 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.

Questions about the ICL method and OOD experiments:
 - 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.
 - 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.
 - 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.


As 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:

E. Hasani and R. A. Ward, “Generating synthetic data for neural operators.” arXiv, Jan. 04, 2024. Available: http://arxiv.org/abs/2401.02398
limitations: limitations: The 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.

Comments: title: Thank you very much for your reply!
Look forward to more discussions
Thank you very much for your reply!
Thank you very much for your reply!
Thank you very much for your reply!
Thank you very much for your reply!
Thank you very much for your reply!
Fix a typo in our response
Thank you very much for your reply!
Response
Continued Responses (2)
Continued Responses (1)
comment: We truly thank reviewer oUb1 for reading our response and providing the reply!

> **Q1**. Model pretrained on vision data in Fig 3a-3d.

Yes, after the rebuttal, we will have enough time to train our own 2D-FNO on ImageNet and a 3D-FNO on video datasets. Then we promise to add vision-trained FNO as another baseline to Poisson, Helmholtz, Reaction-Diffusion, and Navier-Stokes.


> **Q2**. Renaming "in-context learning"

Thanks for this suggestion! We promise to change “ICL” to another name in our camera ready to avoid any confusion.
I thank the authors for the response.

- To support the claim on _even outperforming conventional vision-pretrained models_, I'm still expecting comparisons with a model pretrained on vision data in Fig 3a-3d. The comparisons do not need to be extremely large-scale (e.g., pretraining on JFT-300M).

- I also share the concern of Reviewer oEEz on the effectiveness of the "in-context learning" method. And I still find the claim of "in-context learning" confusing. I agree with Reviewer oEEz that the authors need to focus the contributions on the pretraining strategy. I also suggest that the authors should consider renaming "in-context learning" in the paper as, for example, similarity-based inference with pretrained models.

I will update my rating if the authors can promise to address these concerns in the revision.
Dear Reviewer oUb1,

As the author-reviewer discussion period is nearing its end, and since other reviewers have actively engaged in discussions, we would greatly appreciate it if you could review our responses to your comments at your earliest convenience.

This will allow us to address any further questions or concerns you may have before the discussion period concludes.

If our responses satisfactorily address your concerns, we kindly ask you to consider revising your rating of our work.

Thank you very much for your time and effort!

Sincerely,

The Authors of Submission #11275
I thank again the author for their answer and clarifications. This means that for time dependent PDE, data are needed during the fine-tuning stage in order to learn the temporal dynamics. 
I will raise my score to 6 since all my questions have been answered.
We truly thank reviewer u3i6 for reading our response and providing this further reply!

> **Q1**. How are the mismatched input dimensions handled?

For Helmholtz inputs, we pad extra channels with zeros to match inputs of Poisson.


> **Q2**. When you do the joint pretraining, does the model see all 46,080 unlabeled Poisson samples and all 46,080 unlabeled Helmholtz samples?

* The number of pretraining samples: We used all 46,080 unlabeled Poisson samples and all 46,080 unlabeled Helmholtz samples. We decided to choose this setting because related works on SciML foundation models [1,2] also used all samples from each PDE for pretraining.
* Pretraining costs: We halved the training steps (epochs) to make the total pretraining cost equivalent to pretraining on single PDEs and make the comparison fair.


> **Q3**. The caption of Figure 2.

Thanks for this suggestion! Yes, we will make this caption more precise by noting that they have different distributions of physics parameters in our camera-ready version.


> **Q4**. Simulation costs for the Helmholtz.

Thanks for this suggestion! We are definitely open to further accelerating our simulators. For generating Helmholtz samples, we follow `utils/gen_data_helmholtz.py` in the codebase of [3].

[1] “Multiple Physics Pretraining for Physical Surrogate Models” McCabe et al. 2023

[2] “DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training” Hao et al. 2024.

[3] “Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior“ Subramanian et al. 2023
We truly thank reviewer 6vSy for reading our response and providing this further reply!

For time-dependent PDEs:
* **Pretraining** Phase: neural operators only access individual snapshots, and you are correct: during pretraining, our neural operators only learn spatial reconstruction without learning any temporal dynamics.
* **Fine-tuning** Phase: our neural operators access snapshots with temporal dynamics, and learn both spatial information and forecast the next temporal steps.

We hope this response resolves your further question. We are happy to discuss further!
Thank you very much for the extra information. 

I have a few more questions about the joint pretraining experiment: 
 - How are the mismatched input dimensions handled? As you point out, the Helmholtz and Poisson data have different numbers of channels. How do you train one network to handle both types of data?
 - When you do the joint pretraining, does the model see all 46,080 unlabeled Poisson samples and all 46,080 unlabeled Helmholtz samples? 
 - The caption of Figure 2 in the rebuttal document says Poisson and Helmholtz problems have different initial conditions. Should this say different distributions of physics parameters instead?

The information about the pretraining and simulation costs is encouraging. After seeing these numbers, it is clear that pretraining with your method is a more effective use of compute resources than generating new samples. I believe the paper would be strengthened if this information is included. This was my major concern; I am happy to raise my score as it has been addressed.

As a minor note, I think your estimate for the simulation costs for the Helmholtz equation seems quite high. I believe discretizing the Helmholtz operator + periodic boundary conditions gives a very sparse linear operator which can be inverted quite quickly using standard sparse linear algebra software.
I thank authors for their complete and fast answer to my (numerous) questions. 

-**Q3**: 1. Indeed NO and data-driven method do not have access to physical parameters, however, they have access to the dynamics of the PDE through samples seen during training. From what I understood in your paper, this is not the case of your model, which is unsupervised. This explains my confusion on this point. 
2. Thanks a lot for this table that helps me understand the input.outputs. For static PDE, PDE coefficient and BC could be sufficient to identify the PDE. However, I am confused about time-dependent PDEs: if only ICs are given as input, and that no simulation were proceed, how does the model learns to forecast the next tilmestep?
We truly thank reviewer 6vSy for reading our response and providing the reply!

Below are our responses to your further questions:

> **Q2**. How do you prove these 3 points (experimental).

We would appreciate it if reviewer 6vSy could kindly read our Figure 4 (main submission) and line 292-303. Three subplots in Figure 4 correspond to our three points.


> **Q3**. If your model doesn't have access to the physics and the parameters, how does the model learn the dynamics? Moreover, you state that you model doesn't have access to the physical parameters.

1. In data-driven operator learning, neural operators do not necessarily need to access physical parameters (see [1, 2]). This is mainly because the training PDE data is mostly sampled from a fixed distribution of physical parameters, and the neural operator is trained as a numerical solver for this fixed distribution. This is also why neural operators could perform poorly on out-of-distribution samples.
2. To be more precise, on Reaction-Diffusion and Navier Stokes, FNO and VMAE do not have access to physical parameters. To make this more clear, here we include a table of detailed input/output shapes. Due to the page limit, we will try to squeeze this table into our camera ready.
|  | Input | Input Shape | Output |
|:---:|:---:|:---:|:---:|
| Poisson | source function (f), three diffusion coefficients | CxHxW (C = 4) | potential field (u) |
| Helmholtz | source function (f), wavenumber | CxHxW (C = 2) | wave function (u) |
| NS (FNO) | vorticity (w) | TxHxW (T=33) | vorticity (w) at T+1 |
| NS (PDEBench) | velocity (Vx, Vy), vorticity (w) | TxCxHxW (T=15, C=3) | velocity (Vx, Vy), vorticity (w) at T+1 |
| RD (PDEBench) | activator (u), inhibitor (v) | TxCxHxW (T=10, C=2) | activator (u), inhibitor (v) |


> **Q5**. This behavior is to be expected since the pretrained model have seen more examples than the random init one? Are the examples during pertaining the same as the one used during fine-tuning?

1. This behavior (“random init” is worse than “unsupervised pretrained”) is mainly because of three benefits we discussed in Figure 4: reduced overfitting, faster convergence, and meaningful representations.

2. Examples during pretaining are different from one used during fine-tuning: no overlap.


> **Q8**. Pre training on a single PDE is, I think, very costly, and I would be curious of how long would it take to have a real impact?

Thanks for this question!

Pretraining costs. FNO: 20 GPU hours. Video-MAE: 18 GPU hours. (GPU: A100)

We are working with domain scientists to achieve real impacts on scientific problems as soon as possible (representative examples are climate/weather/airfoil in our Figure 5, where our method has clear improvements).

Meanwhile, we would like to emphasize:
1. Our extra experiments in Figure 1 of our PDF response indicate that **joint unsupervised pretraining on multiple PDEs can reduce the total computational costs**, because fine-tuning on different PDEs can share the same pretrained checkpoint.

2. **Recent works, like [3], also only considered per-PDE pretraining**, even though their work is claimed to target SciML foundation models. Moreover, previous unsupervised pretraining works in computer vision also focused on a single dataset (like ImageNet), and they did not compare the costs of pretraining versus collecting more downstream samples.

We hope the above responses have addressed your concerns. We are happy to discuss this further!


[1] “Multiple Physics Pretraining for Physical Surrogate Models” McCabe et al. 2023

[2] “PDEBENCH: An Extensive Benchmark for Scientific Machine Learning“ Takamoto et al. 2022

[3] “Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior“ Subramanian et al. 2023
I thank the reviewer for their complete answers to my questions. Based on their answers, I still have other questions/clarifications. 

- **Q2**: How do you prove these 3 points (experimental). I think fig. 3 and the OOD experiment answer point n1. But how faster is your method compared to others? Is it in term of computational time or number of steps? 
- **Q3**: I am still not sure to fully understand this point. If your model doesn't have access to the physics and the parameters, how does the model learn the dynamics? Given fixed ICs and/or BCs, several differential operator could be applied to evolve the trajectory. Moreover, you state that you model doesn't have access to the physical parameters. However, on the rebuttal pdf, the legend states that input dimension are different for the reasons of different physical parameters. 
- **Q4**: thanks for the additional experiment, which I believe is the core question for pre training. 
- **Q5** thanks for the clarification. This behavior is to be expected since the *pretrained model* have seen more examples than the *random init* one? Are the examples during pertaining the same as the one used during fine-tuning? 
- **Q8**: I agree with your statement, however, pre training on a single PDE is, I think, very costly, and I would be curious of how long would it take to have a real impact?
We truly thank reviewer oUb1 for reading our response and providing the reply!

Below are our responses to your further questions:

> **Q1**. Pretraining cost and finetuning cost.

1. We reported our pretraining cost in our response to reviewer 6vSy (https://openreview.net/forum?id=MuPlJ9fT4b&noteId=9dXGDW9BRK). We apologize for this omitted response!
  - Pretraining costs. FNO: 20 GPU hours. Video-MAE: 18 GPU hours. (GPU: A100)

2. For fine-tuning costs: FNO: 4 GPU hours. Video-MAE: 6 GPU hours. (GPU: A100).

Meanwhile, we hope that reviewer oUb1 does not ignore our two other responses:
1) **Extra experimental results**: In Figure 1 in our PDF response. We can see that joint pretraining can further improve the performance of fine-tuning on different PDEs. We will include this figure in our camera-ready version.
2) **Previously published works also focused on single-dataset pretraining and did not compare the costs of pretraining/fine-tuning with those of data simulation/collection**: For example, [1] also only adopted per-PDE pretraining, even though their work is claimed to be more related to SciML foundation models than ours. Previous unsupervised pretraining works in computer vision also focused on a single dataset (like ImageNet).


> **Q2**. “no relevant unsupervised learning baseline comparisons in the FNO experiments”

Since we are the first to introduce unsupervised learning in SciML, there are indeed no previous related works (that also train FNO in an unsupervised manner) with which we can fairly compare.


> **Q3**. “Multiple Physics Pretraining for Physical Surrogate Models, show much more comprehensive baseline comparisons in Table 1.”

The reason McCabe et al. needed to compare with more baselines is that their work proposed both a new architecture and pretraining methods. They needed to demonstrate the benefits of both their architecture (by comparing with models with a comparable number of parameters) and their pretraining (by comparing with models trained on a single dataset).

However, in our case, for each subplot in Figure 3: 1) If we compare with different models, we cannot draw meaningful conclusions because the architecture changes; 2) Since ours is the only work on unsupervised pretraining of neural operators, comparing with supervised pretrained models would be unfair.

Instead, we chose the following approach: 1) To address different **architectures**, we studied both FNO and transformer models on different PDEs; 2) To address different **pretraining methods**, we included a vision-pretrained transformer. We believe these two aspects already cover multiple representative baselines that fulfill the purpose requested by reviewer oUb1.


> **Q4**. In-context learning of transformers.

We thank these references, all of them are well awared by we authors. We respect reviewer oUb1’s option. Our definition of “learning” is to update model parameters, and we are open to accepting other definitions.


> **Q5**. Relevant model of billion-level parameters that share similar pretraining strategies as BERT.

One example is the T5 model [2], which is of billion-level parameters and is pretrained with masked language modeling similar to BERT.

Here we quote:
* “we consider an analogous objective to BERT’s “masked language modeling” objective in” in their Sec. 3.
* “we consider an objective inspired by the “masked language modeling” (MLM) objective used in BERT“ in their Sec. 3.3.1.

We again thank reviewer oUb1's reply! We are happy to address any further concerns.




[1] “Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior“ Subramanian et 
al. 2023

[2] “Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer” Raffel et al. 2019
I thank the authors for the detailed rebuttal. I still have several concerns:

- While the authors argue that pretraining on a single PDE are still fundamental and necessary, this increase the computation cost to solve a single PDE. The author should report the total training time of pretraining and finetuning compared to the random initialization experiments for the readers to see the trade-off. I raised questions on the pretraining cost and finetuning cost in my original review but can not find the authors' response on this.

- I thank the authors for trying to add MAE in the FNO experiments. However, at this stage, there is no relevant unsupervised learning baseline comparisons in the FNO experiments. As a reference, the existing work, _Multiple Physics Pretraining for Physical Surrogate Models_, show much more comprehensive baseline comparisons, e.g., in Table 1.

- I disagree with the authors on _models with self-attention are also not learning anything during inference_ - if this is the case, why there should be the notion of "in-context learning"? Existing studies find that Transformers can _learn_ from the few-shot examples even if the weights are fixed [1-4]. This is in contrast to "extracting features" with the proposed model.

[1] Akyürek, Ekin, et al. "What learning algorithm is in-context learning? investigations with linear models." arXiv preprint arXiv:2211.15661 (2022).

[2] Von Oswald, Johannes, et al. "Transformers learn in-context by gradient descent." International Conference on Machine Learning. PMLR, 2023.

[3] Zhang, Ruiqi, Spencer Frei, and Peter L. Bartlett. "Trained transformers learn linear models in-context." arXiv preprint arXiv:2306.09927 (2023).

[4] Bai, Yu, et al. "Transformers as statisticians: Provable in-context learning with in-context algorithm selection." Advances in neural information processing systems 36 (2024).

- The authors promise to _cite more papers with models of billion-level parameters that share similar pretraining strategies as BERT_. I'm not aware of any relevant model like this (i.e., based on masked language modeling). Can the authors kindly share some references?

I maintain my ratings due to the concerns (the first two in particular).
We truly thank reviewer u3i6 for reading our response and providing the prompt reply!

Below are our responses to your further questions:

> **Q1**. Distribution of Helmholtz and Poisson source functions.

Yes, the source distributions are the same for Poisson and Helmholtz. However, they have different ranges of physical parameters in their input, and they have different numbers of input channels (4 for Poisson, 2 for Poisson).


> **Q2**. Pretraining costs and simulation costs.
1. Pretraining costs. FNO: 20 GPU hours. Video-MAE: 18 GPU hours. (GPU: A100)
2. Simulation costs (8192 downstream samples for Figure 3):
* Poisson: 0.04 hour
* Helmholtz: 7.85 hour

> **Q3**. Update new explanations into camera ready.

Yes, we promise to update the explanations for both your previous Q4 and Q6. We thank you for this meaningful discussion!


> **Q4**. Shapes of the pretraining inputs/outputs.

Thanks for the suggestion! We put the shape of our input/output during pretraining into the table below. Since our pretraining is based on reconstruction, the input and output share the same shape.
| PDE | Input/Output (Reconstruction) | Shape |
|---|:---:|:---:|
| Poisson | source function (f), three diffusion coefficients | CxHxW (C = 4) |
| Helmholtz | source function (f), wavenumber | CxHxW (C = 2) |
| NS (FNO) | vorticity (w) | HxW |
| NS (PDEBench) | velocity (Vx, Vy), vorticity (w) | CxHxW (C=3) |
| RD (PDEBench) | activator (u), inhibitor (v) | CxHxW (C=2) |
Many thanks to the authors for providing detailed responses to all parts of the review. I have specific comments and questions about a few parts of the response.

**Q1 and Q9** I really appreciate the design of this extra experiment, and the results look encouraging. However, the lack of detail about the distribution of forcing functions and source functions still obfuscates the significance of these results. I have briefly read [1] to understand the distribution of forcing functions and source functions. To my understanding, the distribution of Helmholtz and Poisson source functions are the same. Is this correct?

**Q2** In my opinion, Figure 3 and the relevant text do not give enough detail to compare the proposed method with the baseline of generating extra samples. This is because there is not enough information to evaluate how long it takes to pretrain the model (I could not find this in Section B3), and there is no information about how long it takes to generate samples for the Helmholtz or Poisson problems. 

**Q4** Thanks for this explanation. Do you plan to update any of the relevant text in the camera-ready version?

**Q6** Thanks for this explanation. The explanation makes sense but is quite different from what appears in the original submission. Do you plan to update this in the camera-ready version?

**Q8** Could you also add the shapes of the pretraining inputs/outputs to this table?

Thank you for all of the other replies; again, I really appreciate the careful and thorough response.
We would like to fix a typo in our response to your "**Q5**. Accuracy and standard deviation of few-shot learning".

"Figure B.3 (f) of [4]" ==> "Figure **B.2** (f) of [4]".
We are referring to subplot (f) in "Figure B.2: Addressing (Q4)" of [4] (in Appendix B.2, at the very bottom of the paper).

[4] “Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior“ Subramanian et al. 2023
We truly thank reviewer oEEz for reading our response and providing the prompt reply!

We understand your concern about our ICL experiments. Meanwhile, we would like to emphasize the following:

1. The **purpose of ICL** is to **continuously reduce the test error with more demos**. This is demonstrated in our figures, where the test error curve drops for Poisson, Helmholtz, and Navier-Stokes. Additionally, we have shown that our method can **scale up to a much larger number of demos** compared to [1].
  - Furthermore, we demonstrate that our ICL method can further reduce the model's uncertainty (evidenced by shorter error bars) as more demos are used. This reduction in uncertainty was not shown in [1].

2. The **value of error** is mainly determined by the **domain gap** between the training source data and the unseen testing data.
Since we chose to test our ICL in an out-of-distribution (OOD) setting, this domain gap is very large (as indicated by the error when #demos = 0).
  - In fact, **our OOD performance is aligned with previous works**, and it is well-known that the performance of neural operators in OOD settings is poor and challenging. For reference, see **subplot (f) in "Figure B.2: Addressing (Q4)" of [2]** (in Appendix B.2, at the very bottom of the paper): Performance on Helmholtz (“SYS-3”) is poor (with errors close to 1 for both their model and ours), even when they fine-tuned with supervision on a few OOD samples.

In sum, we are open to further improving the performance of our ICL method. Meanwhile, we have demonstrated that our ICL method advances beyond previous works [1].

[1] "In-context operator learning with data prompts for differential equation problems" Liu et al. 2023

[2] “Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior“ Subramanian et al. 2023
Thank you for your response.

I believe the pretraining strategy is interesting, especially with the notion of unlabelled data. On the other hand, I am not convinced by the results of the "in-context" learning experiments. While the loss effectively decreases with the number of demo samples, and while using FNO seems to produce better results than with the backbone features (on navier-stokes only), it appears that the gap on Poisson and Helmholtz datasets between in-context and classical experiments is significant (relative errors close to 1 for ICL).  As a result I do not think that the ICL method presented here is effectively taking advantage from the demos.  

I will maintain my score and suggest the authors to only focus their contribution on the pretraining strategy.
> **Q17**. 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?

Collecting snapshots with temporal dynamics in large-scale scenes is significantly more challenging than collecting individual snapshots without temporal dynamics. We have the following examples, and we will try to include them in our camera ready:
1. Weather forecasting: Continuous monitoring of atmospheric parameters requires a network of weather stations, satellites (e.g. large amount of historical satellite data was leveraged in ERA5), and real-time data processing, whereas a single weather report is simpler and less resource-intensive. Long-term climate monitoring involves maintaining observation networks for years, compared to a one-time collection of current climate conditions.
2. Smoke dispersion studies: Tracking smoke spread over time requires multiple sensors (e.g. five cameras with carefully calibrated intrinsics and extrinsics in ScalarFlow) and continuous data processing, unlike a single air quality measurement at a peak pollution event. 
3. Ocean currents: The monitoring demands continuous deployment of buoys and sensors, whereas a single measurement of water conditions can be done with minimal equipment like a handheld sensor or a small research boat.


> **Q18**. Do the authors have any intuition for why the Poisson and Helmholtz OOD performance is poor, and the Navier-Stokes performance is better? 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.

To understand the results in the OOD setting, we provide a visualization in Figure 14 in Appendix K. We find that the Helmholtz equation in the OOD setting exhibits much more complicated unseen patterns than the Poisson and Navier-Stokes equations, making it challenging to learn.

The non-smoothness of the ICL method outputs is mainly due to the uncertainty of neural operators on out-of-distribution (OOD) samples. As the sample-wise similarity is measured by the neural operators’ outputs, which are not accurate, the more severe the OOD setting, the less confident the model’s output and similarity measurement will be.


> **Q19**. 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.

Thanks for this suggestion! 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. We will try to include these examples in our camera ready.
1. 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.
2. 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.
3. 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.


> **Q20**. Regarding related work: E. Hasani and R. A. Ward, “Generating synthetic data for neural operators.” arXiv, Jan. 04, 2024.

Thanks for bringing this up! We noticed this work after our paper submission. Our method, which works on unlabeled PDE data, is orthogonal to their generation of synthetic PDE data, and the two approaches could potentially be combined into a semi-supervised learning strategy.

[1] “Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior“ Subramanian et al. 2023

[2] “SuperBench: A Super-Resolution Benchmark Dataset for Scientific Machine Learning” Ren et al. 2023.

[3] “PDEBENCH: An Extensive Benchmark for Scientific Machine Learning“ Takamoto et al. 2022

[4] “Fourier Neural Operator for Parametric Partial Differential Equations” Li et al. 2020
> **Q8**. 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?

Thanks for this suggestion! Here we include another table of details (input/output shapes). Due to the page limit, we will try to squeeze this table into our camera ready.
|  | Input | Input Shape | Output |
|:---:|:---:|:---:|:---:|
| Poisson | source function (f), three diffusion coefficients | CxHxW (C = 4) | potential field (u) |
| Helmholtz | source function (f), wavenumber | CxHxW (C = 2) | wave function (u) |
| NS (FNO) | vorticity (w) | TxHxW (T=33) | vorticity (w) at T+1 |
| NS (PDEBench) | velocity (Vx, Vy), vorticity (w) | TxCxHxW (T=15, C=3) | velocity (Vx, Vy), vorticity (w) at T+1 |
| RD (PDEBench) | activator (u), inhibitor (v) | TxCxHxW (T=10, C=2) | activator (u), inhibitor (v) |


> **Q9**. For each PDE experiment, what is the distribution of forcing functions (Poisson, Helmholtz) or initial conditions (Reaction-Diffusion, Navier-Stokes)?

For generating forcing functions or initial conditions, we follow previous works [1, 3, 4] and adopt their numerical solvers. We further design the ranges of physical parameters as listed in Table 1.


> **Q10**. 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?

The "diffusion" physics parameter in Table 1 determines the eigenvalue of the diffusion coefficient tensor. For more details about the construction of diffusion coefficient tensors, please consider reading the paragraph “PDE coefficient sampling” in Section 3 of [1].


> **Q11**. How does the Reynolds number relate to the definition of the Navier-Stokes equations (Equation 5)?

The Reynolds number is defined as $Re=\frac{\rho u L}{\nu}$, where $\nu$ is the viscosity of the fluid ($\rho$ is the density of the fluid, $u$ is the flow speed, $L$ is a constant of the fluid’s characteristic linear dimension). $\nu$ is used in Eq. 5. In other words, a smaller $\nu$ indicates a larger Reynolds number.


> **Q12**. Are there spatial boundary conditions for the Reaction-Diffusion equation?

Following PDEBench [3], we consider the Neumann boundary condition for Reaction-Diffusion.


> **Q13**. What do the error bars in Figures 3 and 4 represent?

Error bars in Figures 3 and 4 indicate standard deviations of three independent runs with different random seeds.


> **Q14**. 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?

Similar to time-dependent PDEs (reaction-diffusion, Navier-Stokes), which are explained in line 131 and in Appendix A, on ERA5 and ScalarFlow, models are trained on individual snapshots without temporal information. The pretraining, fine-tuning, and test sets are separate without overlap. We will provide clearer explanations in our camera-ready version.


> **Q15**. 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?

In time-dependent PDEs, the simulation can start from any snapshot (i.e., the numerical simulation can take any snapshot as the initial condition and continue simulating the dynamics). In fact, in FNO, the actual simulation of Navier-Stokes starts after the fluid roughly exits the chaotic phases (please refer to the GitHub repo called `neuraloperator/physics_informed`, line 34 of `generate_data.py`).

This setting is also similar to starting the collection of climate data at any timestep and continuing the collection. The key is that we do not involve any temporal dynamics during our pretraining.


> **Q16**. 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?

The “random init” baseline can access this data. However, as we consider these pretraining samples as individual snapshots (to simulate the practice where people cannot capture temporal dynamics and can only collect individual spatial climate/flow snapshots), we do not use any temporal information in these samples, which cannot be used for supervised forecasting.