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In this paper, we propose new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher order accuracy in space and time. As a model problem, the convection-diffusion problem on a moving domain is studied. For geometrically higher order accuracy, we apply a parametric mapping on a background space-time tensor-product mesh. Concerning discretisation in time, we consider discontinuous Galerkin, as well as related continuous (Petrov-)Galerkin and Galerkin collocation methods. For stabilisation with respect to bad cut configurations and as an extension mechanism that is required for the latter two schemes, a ghost penalty stabilisation is employed. The article puts an emphasis on the techniques that allow to achieve a robust but higher order geometry handling for smooth domains. We investigate the computational properties of the respective methods in a series of numerical experiments. These include studies in different dimensions for different polynomial degrees in space and time, validating the higher order accuracy in both variables.

This paper investigates the problem of learning robust, generalizable prediction models from a combination of multiple datasets and qualitative assumptions about the underlying data-generating model. Part of the challenge of learning robust models lies in the influence of unobserved confounders that void many of the invariances and principles of minimum error presently used for this problem. Our approach is to define a different invariance property of causal solutions in the presence of unobserved confounders which, through a relaxation of this invariance, can be connected with an explicit distributionally robust optimization problem over a set of affine combination of data distributions. Concretely, our objective takes the form of a standard loss, plus a regularization term that encourages partial equality of error derivatives with respect to model parameters. We demonstrate the empirical performance of our approach on healthcare data from different modalities, including image, speech and tabular data.

We consider a singularly perturbed time dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Dur\'an- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

We revisit Domain Generalization (DG) problem, where the hypotheses are composed of a common representation mapping followed by a labeling function. Popular DG methods optimize a well-known upper bound to the risk in the unseen domain. However, the bound contains a term that is not optimized due to its dual dependence on the representation mapping and the unknown optimal labeling function for the unseen domain. We derive a new upper bound free of terms having such dual dependence by imposing mild assumptions on the loss function and an invertibility requirement on the representation map when restricted to the low-dimensional data manifold. The derivation leverages old and recent transport inequalities that link optimal transport metrics with information-theoretic measures. Our bound motivates a new algorithm for DG comprising Wasserstein-2 barycenter cost for feature alignment and mutual information or autoencoders for enforcing approximate invertibility. Experiments on several datasets demonstrate superior performance compared to the state-of-the-art DG algorithms.

The distributed convex optimization problem over the multi-agent system is considered in this paper, and it is assumed that each agent possesses its own cost function and communicates with its neighbours over a sequence of time-varying directed graphs. However, due to some reasons there exist communication delays while agents receive information from other agents, and we are going to seek the optimal value of the sum of agents' loss functions in this case. We desire to handle this problem with the push-sum distributed dual averaging (PS-DDA) algorithm. It is proved that this algorithm converges and the error decays at a rate $\mathcal{O}\left(T^{-0.5}\right)$ with proper step size, where $T$ is iteration span. The main result presented in this paper also illustrates the convergence of the proposed algorithm is related to the maximum value of the communication delay on one edge. We finally apply the theoretical results to numerical simulations to show the PS-DDA algorithm's performance.

Mainstream approaches for unsupervised domain adaptation (UDA) learn domain-invariant representations to narrow the domain shift. Recently, self-training has been gaining momentum in UDA, which exploits unlabeled target data by training with target pseudo-labels. However, as corroborated in this work, under distributional shift in UDA, the pseudo-labels can be unreliable in terms of their large discrepancy from target ground truth. Thereby, we propose Cycle Self-Training (CST), a principled self-training algorithm that explicitly enforces pseudo-labels to generalize across domains. CST cycles between a forward step and a reverse step until convergence. In the forward step, CST generates target pseudo-labels with a source-trained classifier. In the reverse step, CST trains a target classifier using target pseudo-labels, and then updates the shared representations to make the target classifier perform well on the source data. We introduce the Tsallis entropy as a confidence-friendly regularization to improve the quality of target pseudo-labels. We analyze CST theoretically under realistic assumptions, and provide hard cases where CST recovers target ground truth, while both invariant feature learning and vanilla self-training fail. Empirical results indicate that CST significantly improves over the state-of-the-arts on visual recognition and sentiment analysis benchmarks.

Domain generalization aims to learn a generalizable model from a known source domain for various unknown target domains. It has been studied widely by domain randomization that transfers source images to different styles in spatial space for learning domain-agnostic features. However, most existing randomization uses GANs that often lack of controls and even alter semantic structures of images undesirably. Inspired by the idea of JPEG that converts spatial images into multiple frequency components (FCs), we propose Frequency Space Domain Randomization (FSDR) that randomizes images in frequency space by keeping domain-invariant FCs (DIFs) and randomizing domain-variant FCs (DVFs) only. FSDR has two unique features: 1) it decomposes images into DIFs and DVFs which allows explicit access and manipulation of them and more controllable randomization; 2) it has minimal effects on semantic structures of images and domain-invariant features. We examined domain variance and invariance property of FCs statistically and designed a network that can identify and fuse DIFs and DVFs dynamically through iterative learning. Extensive experiments over multiple domain generalizable segmentation tasks show that FSDR achieves superior segmentation and its performance is even on par with domain adaptation methods that access target data in training.

Existing domain adaptation focuses on transferring knowledge between domains with categorical indices (e.g., between datasets A and B). However, many tasks involve continuously indexed domains. For example, in medical applications, one often needs to transfer disease analysis and prediction across patients of different ages, where age acts as a continuous domain index. Such tasks are challenging for prior domain adaptation methods since they ignore the underlying relation among domains. In this paper, we propose the first method for continuously indexed domain adaptation. Our approach combines traditional adversarial adaptation with a novel discriminator that models the encoding-conditioned domain index distribution. Our theoretical analysis demonstrates the value of leveraging the domain index to generate invariant features across a continuous range of domains. Our empirical results show that our approach outperforms the state-of-the-art domain adaption methods on both synthetic and real-world medical datasets.

Recent work has shown that a variety of semantics emerge in the latent space of Generative Adversarial Networks (GANs) when being trained to synthesize images. However, it is difficult to use these learned semantics for real image editing. A common practice of feeding a real image to a trained GAN generator is to invert it back to a latent code. However, existing inversion methods typically focus on reconstructing the target image by pixel values yet fail to land the inverted code in the semantic domain of the original latent space. As a result, the reconstructed image cannot well support semantic editing through varying the inverted code. To solve this problem, we propose an in-domain GAN inversion approach, which not only faithfully reconstructs the input image but also ensures the inverted code to be semantically meaningful for editing. We first learn a novel domain-guided encoder to project a given image to the native latent space of GANs. We then propose domain-regularized optimization by involving the encoder as a regularizer to fine-tune the code produced by the encoder and better recover the target image. Extensive experiments suggest that our inversion method achieves satisfying real image reconstruction and more importantly facilitates various image editing tasks, significantly outperforming start-of-the-arts.

The spatial convolution layer which is widely used in the Graph Neural Networks (GNNs) aggregates the feature vector of each node with the feature vectors of its neighboring nodes. The GNN is not aware of the locations of the nodes in the global structure of the graph and when the local structures corresponding to different nodes are similar to each other, the convolution layer maps all those nodes to similar or same feature vectors in the continuous feature space. Therefore, the GNN cannot distinguish two graphs if their difference is not in their local structures. In addition, when the nodes are not labeled/attributed the convolution layers can fail to distinguish even different local structures. In this paper, we propose an effective solution to address this problem of the GNNs. The proposed approach leverages a spatial representation of the graph which makes the neural network aware of the differences between the nodes and also their locations in the graph. The spatial representation which is equivalent to a point-cloud representation of the graph is obtained by a graph embedding method. Using the proposed approach, the local feature extractor of the GNN distinguishes similar local structures in different locations of the graph and the GNN infers the topological structure of the graph from the spatial distribution of the locally extracted feature vectors. Moreover, the spatial representation is utilized to simplify the graph down-sampling problem. A new graph pooling method is proposed and it is shown that the proposed pooling method achieves competitive or better results in comparison with the state-of-the-art methods.