Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery

Summary

This work presents a new neural network for tackling PDEs' forward and backward problems, especially for small physics systems. The forward problem simulates the outcome of a physics system given the inputs, while the backward problem estimates the parameters of a physics system given the inputs and outputs. The key idea of the presented method is to utilize the underlying common rules of multiple physics systems. This work found that using an attention layer to model multiple physics systems can significantly help the model solve the backward problem, also known as the discovery problem. The idea is intuitive: looking at multiple similar rules is helpful for discovering a new rule.

Strengths

The idea is interesting and clever. The nature of sharing knowledge among multiple systems is somewhat related to in-context learning in LLMs.

Weaknesses

The experiments could be more extensive and solid, especially in supporting the main contribution. Although the experiment on the backward problem (discovery) reports the OOD results, the OOD setting is not far from the in-domain data, which raises concerns about the generalization ability. The paper's writing is not very friendly to those with a deep learning background. The content around background knowledge and the connections between the attention mechanism and kernel methods is somewhat tangential to the main storyline. The idea is straightforward, so I would prefer a clean and compact narrative. However, this is not a major concern since I only have a deep learning background.

Questions

For Section 5.1, consider including a broader range of tasks and different types of kernels. It would also be beneficial to incorporate more OOD testing. Another related question is: Given that multiple training tasks come from the same family, could this be equivalent to a weaker version of learning more data pairs from a single system? I am unsure whether the reported number of data pairs refers to the total data pairs across different tasks or for each individual task; I suspect it is the latter. If we provide more data pairs from a single task, could it achieve a similar level of performance? Conversely, if we increase the number of tasks and reduce the data pairs for each task, does this improve the generalization ability?

Rating

7

Confidence

3

Soundness

2

Presentation

2

Contribution

3

Limitations

Agree with the limitation section.

Summary

This paper explores using large-language deep-neural network to build a foundation physical model that can solve inverse PDE problem. It proposes Nonlocal Attention Operator model with a kernel map that is dependent on both input and output functions. The intermediate proofs help people understand the underlying math principles behind the attention mechanism, especially the relationship between attention and integration. The paper also does experiments on radial kernel learning, solution operator learning and heterogeneous material learning, all with decent results.

Strengths

- mathematical proof is comprehensive - ablation study has compared performance among several variants - experiments are adequate to show the effectiveness of the model in different domains

Weaknesses

- there are very few comparisons with other papers on experimental results

Questions

- is it possible for you publish your source codes on Github/Gitlab so that other people can use your framework for practical applications and researchers can expand on top of your work?

Rating

7

Confidence

2

Soundness

3

Presentation

3

Contribution

3

Limitations

- since the equations are quite complex, replication of the piece of work might not be consistent among different people, thus, lack of open-source code will limit its use/adoption in practical applications.

Summary

This paper introduces a novel approach called the Nonlocal Attention Operator (NAO) for simultaneously solving forward and inverse PDE problems across multiple physical systems. The key innovation is using an attention-based kernel map that learns to infer resolution-invariant kernels from datasets, enabling both physics modeling and mechanism discovery. Thank you to the authors for their responses. It is great to see a detailed complexity analysis; still, the quadratic scaling is definitely a cause for concern -- hence the reasons for relaxing kernel-based formulations of self-attention to linear scaling versions (e.g., Mamba, informer, etc.). However, I can see how similar relaxations can be applied to NAO. So this is very helpful.

Strengths

The main strengths I find are as follows: (i) Bridging forward and inverse PDE modeling: The approach uniquely addresses forward PDE solving (prediction) and inverse PDE solving (discovering hidden mechanisms) in a unified framework. (ii) Interpretability: The discovered kernel provides meaningful physical interpretation, addressing a common criticism of black-box neural network approaches. (iii) Theoretical foundation: The authors provide a theoretical analysis showing how the attention mechanism provides a space of identifiability for kernels, helping to resolve ill-posed inverse problems. The authors also provide, as part of their approach, extensibility to various forms of operators, including those with radial interaction kernels and heterogeneous interactions. (iv) Impact: Novel mechanisms for modeling and learning about physical phenomena with a focus on interpretability is a significant step in the right direction for me. I am unsure that this will directly lead to contributions in the foundation model space (I think it's a bit of a stretch -- see weaknesses for details). However, I can see that this work can be gainfully leveraged to drive insights for foundation model building. My reading of the analysis provided also yielded the following strengths (specifically regarding the formal rigor of the proposed approach): The authors present a general form of the operator in Eq. (3), which encompasses many physical systems. This generality is a strength of the approach. The kernel map (Eq. 4-7) uses a multi-layer attention model to learn the mapping from data pairs to kernel estimations. The use of both input (u) and output (f) in the attention mechanism is novel and crucial for addressing the inverse problem. The derivation of a continuum limit of the discrete attention model (Eq. 8) provides insight into the operator's behavior as the number of data points increases. Lemmas 4.1 and 4.2 provide valuable insights into the limit behavior of the attention-parameterized kernel and the space of identifiability for kernels. This analysis helps to explain why the method can effectively solve ill-posed inverse problems.

Weaknesses

I find only one glaring weakness, which is likely to surface during the implementation of this approach for physical systems of appreciable size and complexity: The method involves multiple integrations and matrix operations, which may be computationally expensive for large-scale problems. A discussion of computational efficiency and scalability would be beneficial. On a related note, It is also not clear to me if this method is data-efficient. A minor weakness I see is that I do not find that this method is easily integrated with current approaches to physics-informed machine learning, e.g., through simple extensions to pre-existing libraries. A brief discussion on this very practical aspect might strengthen the paper.

Questions

1. How does the computational complexity of NAO scale with the number of data points and dimensionality of the problem? Provide a detailed analysis of computational costs and compare them with existing methods for both forward and inverse PDE solving. 2. What are the minimum data requirements for NAO to perform effectively, particularly for the inverse problem? It may be beneficial to provide an analysis of performance vs. data quantity to guide potential users in determining data collection needs. 3. How can NAO be integrated with existing physics-based models or simulations to enhance their capabilities? Maybe discuss potential hybrid approaches combining NAO with existing or traditional methods.

Rating

7

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

I read the limitation sections, and the authors provide, I think, an inkling of the answers to the questions that I have raised. I hope to see a more comprehensive discussion on the same, with the necessary addendums made to the limitations section.

Summary

This paper proposes a novel neural operator architecture called Nonlocal Attention Operator (NAO) for learning both forward and inverse problems in physical systems from data. The key contributions are: 1. A new attention-based neural operator that can simultaneously learn the forward mapping (physics modeling) and inverse mapping (physics discovery) for PDE systems. 2. Theoretical analysis showing how the attention mechanism provides a kernel map that explores the space of identifiability for kernels, helping to resolve ill-posedness in inverse problems. 3. Empirical demonstration of NAO's advantages over baseline methods, especially for generalization to unseen systems and data-efficient learning in ill-posed inverse problems. 4. Interpretability of the learned kernels, allowing insight into the discovered physical mechanisms. The authors evaluate NAO on several PDE learning tasks, including radial kernel learning, solution operator learning for Darcy flow, and heterogeneous nonlinear material modeling.

Strengths

1. The paper presents a novel neural operator architecture that leverages attention in a unique way for PDE learning. The idea of using attention to build a kernel map that can generalize across multiple physical systems is innovative. 2. The theoretical analysis in Section 4 provides rigorous justification for how the attention mechanism helps address ill-posedness. The empirical evaluations are comprehensive, testing multiple aspects like generalization, data efficiency, and interpretability. 3. The paper is generally well-written and clearly structured. The motivation, methodology, and results are presented logically. 4. The ability to simultaneously learn forward and inverse mappings for PDE systems, while providing interpretable kernels, could have broad impact in scientific machine learning and physics-informed AI. The zero-shot generalization capability is particularly notable.

Weaknesses

1. The experimental evaluation is limited to relatively simple PDE systems. It's unclear how well NAO would scale to more complex, high-dimensional PDEs encountered in real-world applications. 2. The interpretability claims could be further substantiated. While learned kernels are visualized, there's limited discussion on how these provide physical insights beyond matching ground truth.

Questions

Can you provide more insight into how the learned kernels can be interpreted to gain physical understanding? Perhaps a case study showing how NAO discovers meaningful structure in a physical system would be illuminating.

Rating

7

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

Yes.

Response

Thanks for the detailed response and new experiment results. I will raise my score.

response to rebuttal

I have read the rebuttal. Thanks for your work!