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Benjamin J. Zhang, Siting Liu, Wuchen Li, Markos A. Katsoulakis, Stanley J. Osher

We focus on the fundamental mathematical structure of score-based generative models (SGMs). We first formulate SGMs in terms of the Wasserstein proximal operator (WPO) and demonstrate that, via mean-field games (MFGs), the WPO formulation reveals mathematical structure that describes the inductive bias of diffusion and score-based models. In particular, MFGs yield optimality conditions in the form of a pair of coupled partial differential equations: a forward-controlled Fokker-Planck (FP) equation, and a backward Hamilton-Jacobi-Bellman (HJB) equation. Via a Cole-Hopf transformation and taking advantage of the fact that the cross-entropy can be related to a linear functional of the density, we show that the HJB equation is an uncontrolled FP equation. Second, with the mathematical structure at hand, we present an interpretable kernel-based model for the score function which dramatically improves the performance of SGMs in terms of training samples and training time. In addition, the WPO-informed kernel model is explicitly constructed to avoid the recently studied memorization effects of score-based generative models. The mathematical form of the new kernel-based models in combination with the use of the terminal condition of the MFG reveals new explanations for the manifold learning and generalization properties of SGMs, and provides a resolution to their memorization effects. Finally, our mathematically informed, interpretable kernel-based model suggests new scalable bespoke neural network architectures for high-dimensional applications.

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Liu Yang, Stanley J. Osher

Can we build a single large model for a wide range of PDE-related scientific learning tasks? Can this model generalize to new PDEs, even of new forms, without any fine-tuning? In-context operator learning and the corresponding model In-Context Operator Networks (ICON) represent an initial exploration of these questions. The capability of ICON regarding the first question has been demonstrated previously. In this paper, we present a detailed methodology for solving PDE problems with ICON, and show how a single ICON model can make forward and reverse predictions for different equations with different strides, provided with appropriately designed data prompts. We show the positive evidence to the second question, i.e., ICON can generalize well to some PDEs with new forms without any fine-tuning. This is exemplified through a study on 1D scalar nonlinear conservation laws, a family of PDEs with temporal evolution. We also show how to broaden the range of problems that an ICON model can address, by transforming functions and equations to ICON's capability scope. We believe that the progress in this paper is a significant step towards the goal of training a foundation model for PDE-related tasks under the in-context operator learning framework.

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Liu Yang, Tingwei Meng, Siting Liu, Stanley J. Osher

In the growing domain of scientific machine learning, in-context operator learning has demonstrated notable potential in learning operators from prompted data during inference stage without weight updates. However, the current model's overdependence on sensor data, may inadvertently overlook the invaluable human insight into the operator. To address this, we present a transformation of in-context operator learning into a multi-modal paradigm. We propose the use of "captions" to integrate human knowledge about the operator, expressed through natural language descriptions and equations. We illustrate how this method not only broadens the flexibility and generality of physics-informed learning, but also significantly boosts learning performance and reduces data needs. Furthermore, we introduce a more efficient neural network architecture for multi-modal in-context operator learning, referred to as "ICON-LM", based on a language-model-like architecture. We demonstrate the viability of "ICON-LM" for scientific machine learning tasks, which creates a new path for the application of language models.

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Liu Yang, Siting Liu, Tingwei Meng, Stanley J. Osher

This paper introduces a new neural-network-based approach, namely IN-context Differential Equation Encoder-Decoder (INDEED), to simultaneously learn operators from data and apply it to new questions during the inference stage, without any weight update. Existing methods are limited to using a neural network to approximate a specific equation solution or a specific operator, requiring retraining when switching to a new problem with different equations. By training a single neural network as an operator learner, we can not only get rid of retraining (even fine-tuning) the neural network for new problems, but also leverage the commonalities shared across operators so that only a few demos are needed when learning a new operator. Our numerical results show the neural network's capability as a few-shot operator learner for a diversified type of differential equation problems, including forward and inverse problems of ODEs and PDEs, and also show that it can generalize its learning capability to operators beyond the training distribution, even to an unseen type of operator.

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Tan Nguyen, Richard G. Baraniuk, Robert M. Kirby, Stanley J. Osher, Bao Wang

Transformers have achieved remarkable success in sequence modeling and beyond but suffer from quadratic computational and memory complexities with respect to the length of the input sequence. Leveraging techniques include sparse and linear attention and hashing tricks; efficient transformers have been proposed to reduce the quadratic complexity of transformers but significantly degrade the accuracy. In response, we first interpret the linear attention and residual connections in computing the attention map as gradient descent steps. We then introduce momentum into these components and propose the \emph{momentum transformer}, which utilizes momentum to improve the accuracy of linear transformers while maintaining linear memory and computational complexities. Furthermore, we develop an adaptive strategy to compute the momentum value for our model based on the optimal momentum for quadratic optimization. This adaptive momentum eliminates the need to search for the optimal momentum value and further enhances the performance of the momentum transformer. A range of experiments on both autoregressive and non-autoregressive tasks, including image generation and machine translation, demonstrate that the momentum transformer outperforms popular linear transformers in training efficiency and accuracy.

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Alex Tong Lin, Stanley J. Osher

We introduce a method called MASCOT (Multi-Agent Shape Control with Optimal Transport) to compute optimal control solutions of agents with shape/formation/density constraints. For example, we might want to apply shape constraints on the agents -- perhaps we desire the agents to hold a particular shape along the path, or we want agents to spread out in order to minimize collisions. We might also want a proportion of agents to move to one destination, while the other agents move to another, and to do this in the optimal way, i.e. the source-destination assignments should be optimal. In order to achieve this, we utilize the Earth Mover's Distance from Optimal Transport to distribute the agents into their proper positions so that certain shapes can be satisfied. This cost is both introduced in the terminal cost and in the running cost of the optimal control problem.

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Tan Nguyen, Minh Pham, Tam Nguyen, Khai Nguyen, Stanley J. Osher, Nhat Ho

Multi-head attention empowers the recent success of transformers, the state-of-the-art models that have achieved remarkable success in sequence modeling and beyond. These attention mechanisms compute the pairwise dot products between the queries and keys, which results from the use of unnormalized Gaussian kernels with the assumption that the queries follow a mixture of Gaussian distribution. There is no guarantee that this assumption is valid in practice. In response, we first interpret attention in transformers as a nonparametric kernel regression. We then propose the FourierFormer, a new class of transformers in which the dot-product kernels are replaced by the novel generalized Fourier integral kernels. Different from the dot-product kernels, where we need to choose a good covariance matrix to capture the dependency of the features of data, the generalized Fourier integral kernels can automatically capture such dependency and remove the need to tune the covariance matrix. We theoretically prove that our proposed Fourier integral kernels can efficiently approximate any key and query distributions. Compared to the conventional transformers with dot-product attention, FourierFormers attain better accuracy and reduce the redundancy between attention heads. We empirically corroborate the advantages of FourierFormers over the baseline transformers in a variety of practical applications including language modeling and image classification.

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Justin Baker, Hedi Xia, Yiwei Wang, Elena Cherkaev, Akil Narayan, Long Chen, Jack Xin, Andrea L. Bertozzi, Stanley J. Osher, Bao Wang

Learning neural ODEs often requires solving very stiff ODE systems, primarily using explicit adaptive step size ODE solvers. These solvers are computationally expensive, requiring the use of tiny step sizes for numerical stability and accuracy guarantees. This paper considers learning neural ODEs using implicit ODE solvers of different orders leveraging proximal operators. The proximal implicit solver consists of inner-outer iterations: the inner iterations approximate each implicit update step using a fast optimization algorithm, and the outer iterations solve the ODE system over time. The proximal implicit ODE solver guarantees superiority over explicit solvers in numerical stability and computational efficiency. We validate the advantages of proximal implicit solvers over existing popular neural ODE solvers on various challenging benchmark tasks, including learning continuous-depth graph neural networks and continuous normalizing flows.

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Tam Nguyen, Tan M. Nguyen, Dung Le, Khuong Nguyen, Anh Tran, Richard G. Baraniuk, Nhat Ho, Stanley J. Osher

Multi-head attention is a driving force behind state-of-the-art transformers which achieve remarkable performance across a variety of natural language processing (NLP) and computer vision tasks. It has been observed that for many applications, those attention heads learn redundant embedding, and most of them can be removed without degrading the performance of the model. Inspired by this observation, we propose Transformer with a Mixture of Gaussian Keys (Transformer-MGK), a novel transformer architecture that replaces redundant heads in transformers with a mixture of keys at each head. These mixtures of keys follow a Gaussian mixture model and allow each attention head to focus on different parts of the input sequence efficiently. Compared to its conventional transformer counterpart, Transformer-MGK accelerates training and inference, has fewer parameters, and requires less FLOPs to compute while achieving comparable or better accuracy across tasks. Transformer-MGK can also be easily extended to use with linear attentions. We empirically demonstrate the advantage of Transformer-MGK in a range of practical applications including language modeling and tasks that involve very long sequences. On the Wikitext-103 and Long Range Arena benchmark, Transformer-MGKs with 4 heads attain comparable or better performance to the baseline transformers with 8 heads.

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