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Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other deep learning methods. However, the spatial domain of its input function needs to be identical to its output, which limits its applicability. For instance, the widely used Fourier neural operator (FNO) fails to approximate the operator that maps the boundary condition to the PDE solution. To address this issue, we propose a novel framework called resolution-invariant deep operator (RDO) that decouples the spatial domain of the input and output. RDO is motivated by the Deep operator network (DeepONet) and it does not require retraining the network when the input/output is changed compared with DeepONet. RDO takes functional input and its output is also functional so that it keeps the resolution invariant property of NO. It can also resolve PDEs with complex geometries whereas NO fail. Various numerical experiments demonstrate the advantage of our method over DeepONet and FNO.

A mesh motion model based on deep operator networks is presented. The model is trained on and evaluated against a biharmonic mesh motion model on a fluid-structure interaction benchmark problem and further evaluated in a setting where biharmonic mesh motion fails. The performance of the proposed mesh motion model is comparable to the biharmonic mesh motion on the test problems.

Revealing hidden dynamics from the stochastic data is a challenging problem as randomness takes part in the evolution of the data. The problem becomes exceedingly complex when the trajectories of the stochastic data are absent in many scenarios. Here we present an approach to effectively modeling the dynamics of the stochastic data without trajectories based on the weak form of the Fokker-Planck (FP) equation, which governs the evolution of the density function in the Brownian process. Taking the collocations of Gaussian functions as the test functions in the weak form of the FP equation, we transfer the derivatives to the Gaussian functions and thus approximate the weak form by the expectational sum of the data. With a dictionary representation of the unknown terms, a linear system is built and then solved by the regression, revealing the unknown dynamics of the data. Hence, we name the method with the Weak Collocation Regression (WCR) method for its three key components: weak form, collocation of Gaussian kernels, and regression. The numerical experiments show that our method is flexible and fast, which reveals the dynamics within seconds in multi-dimensional problems and can be easily extended to high-dimensional data such as 20 dimensions. WCR can also correctly identify the hidden dynamics of the complex tasks with variable-dependent diffusion and coupled drift, and the performance is robust, achieving high accuracy in the case with noise added.

Despite substantial efforts, neural network interpretability remains an elusive goal, with previous research failing to provide succinct explanations of most single neurons' impact on the network output. This limitation is due to the polysemantic nature of most neurons, whereby a given neuron is involved in multiple unrelated network states, complicating the interpretation of that neuron. In this paper, we apply tools developed in neuroscience and information theory to propose both a novel practical approach to network interpretability and theoretical insights into polysemanticity and the density of codes. We infer levels of redundancy in the network's code by inspecting the eigenspectrum of the activation's covariance matrix. Furthermore, we show how random projections can reveal whether a network exhibits a smooth or non-differentiable code and hence how interpretable the code is. This same framework explains the advantages of polysemantic neurons to learning performance and explains trends found in recent results by Elhage et al.~(2022). Our approach advances the pursuit of interpretability in neural networks, providing insights into their underlying structure and suggesting new avenues for circuit-level interpretability.

Aqueous solubility is a valuable yet challenging property to predict. Computing solubility using first-principles methods requires accounting for the competing effects of entropy and enthalpy, resulting in long computations for relatively poor accuracy. Data-driven approaches, such as deep learning, offer improved accuracy and computational efficiency but typically lack uncertainty quantification. Additionally, ease of use remains a concern for any computational technique, resulting in the sustained popularity of group-based contribution methods. In this work, we addressed these problems with a deep learning model with predictive uncertainty that runs on a static website (without a server). This approach moves computing needs onto the website visitor without requiring installation, removing the need to pay for and maintain servers. Our model achieves satisfactory results in solubility prediction. Furthermore, we demonstrate how to create molecular property prediction models that balance uncertainty and ease of use. The code is available at //github.com/ur-whitelab/mol.dev, and the model is usable at //mol.dev.

We consider the problem of estimating the marginal independence structure of a Bayesian network from observational data, learning an undirected graph we call the unconditional dependence graph. We show that unconditional dependence graphs of Bayesian networks correspond to the graphs having equal independence and intersection numbers. Using this observation, a Gr\"obner basis for a toric ideal associated to unconditional dependence graphs of Bayesian networks is given and then extended by additional binomial relations to connect the space of all such graphs. An MCMC method, called GrUES (Gr\"obner-based Unconditional Equivalence Search), is implemented based on the resulting moves and applied to synthetic Gaussian data. GrUES recovers the true marginal independence structure via a penalized maximum likelihood or MAP estimate at a higher rate than simple independence tests while also yielding an estimate of the posterior, for which the $20\%$ HPD credible sets include the true structure at a high rate for data-generating graphs with density at least $0.5$.

In this paper, a deep learning method for solving an improved one-dimensional Poisson-Nernst-Planck ion channel (PNPic) model, called the PNPic deep learning solver, is presented. In particular, it combines a novel local neural network scheme with an effective PNPic finite element solver. Since the input data of the neural network scheme only involves a small local patch of coarse grid solutions, which the finite element solver can quickly produce, the PNPic deep learning solver can be trained much faster than any corresponding conventional global neural network solvers. After properly trained, it can output a predicted PNPic solution in a much higher degree of accuracy than the low cost coarse grid solutions and can reflect different perturbation cases on the parameters, ion channel subregions, and interface and boundary values, etc. Consequently, the PNPic deep learning solver can generate a numerical solution with high accuracy for a family of PNPic models. As an initial study, two types of numerical tests were done by perturbing one and two parameters of the PNPic model, respectively, as well as the tests done by using a few perturbed interface positions of the model as training samples. These tests demonstrate that the PNPic deep learning solver can generate highly accurate PNPic numerical solutions.

We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.