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We propose a novel formalism for describing Structural Causal Models (SCMs) as fixed-point problems on causally ordered variables, eliminating the need for Directed Acyclic Graphs (DAGs), and establish the weakest known conditions for their unique recovery given the topological ordering (TO). Based on this, we design a two-stage causal generative model that first infers in a zero-shot manner a valid TO from observations, and then learns the generative SCM on the ordered variables. To infer TOs, we propose to amortize the learning of TOs on synthetically generated datasets by sequentially predicting the leaves of graphs seen during training. To learn SCMs, we design a transformer-based architecture that exploits a new attention mechanism enabling the modeling of causal structures, and show that this parameterization is consistent with our formalism. Finally, we conduct an extensive evaluation of each method individually, and show that when combined, our model outperforms various baselines on generated out-of-distribution problems. The code is available on \href{//github.com/microsoft/causica/tree/main/research_experiments/fip}{Github}.

In this paper, we study the multi-patch discontinuous Galerkin isogeometric (DG-IGA) approximations for full-potential electronic structure calculations. We decompose the physical domain into several subdomains, represent each part of the wavefunction separately using B-spline basis functions, possibly with different degrees, on varying mesh sizes, and then combine them by DG methods. We also provide a rigorous {\em a priori} error analysis of the DG-IGA approximations for linear eigenvalue problems. Furthermore, this work offers a unified analysis framework for the DG-IGA method applied to a class of elliptic eigenvalue problems. Finally, we present several numerical experiments to verify our theoretical results.

Gaussian Process differential equations (GPODE) have recently gained momentum due to their ability to capture dynamics behavior of systems and also represent uncertainty in predictions. Prior work has described the process of training the hyperparameters and, thereby, calibrating GPODE to data. How to design efficient algorithms to collect data for training GPODE models is still an open field of research. Nevertheless high-quality training data is key for model performance. Furthermore, data collection leads to time-cost and financial-cost and might in some areas even be safety critical to the system under test. Therefore, algorithms for safe and efficient data collection are central for building high quality GPODE models. Our novel Safe Active Learning (SAL) for GPODE algorithm addresses this challenge by suggesting a mechanism to propose efficient and non-safety-critical data to collect. SAL GPODE does so by sequentially suggesting new data, measuring it and updating the GPODE model with the new data. In this way, subsequent data points are iteratively suggested. The core of our SAL GPODE algorithm is a constrained optimization problem maximizing information of new data for GPODE model training constrained by the safety of the underlying system. We demonstrate our novel SAL GPODE's superiority compared to a standard, non-active way of measuring new data on two relevant examples.

Large language models (LLMs) have exhibited remarkable few-shot learning capabilities and unified the paradigm of NLP tasks through the in-context learning (ICL) technique. Despite the success of ICL, the quality of the exemplar demonstrations can significantly influence the LLM's performance. Existing exemplar selection methods mainly focus on the semantic similarity between queries and candidate exemplars. On the other hand, the logical connections between reasoning steps can be beneficial to depict the problem-solving process as well. In this paper, we proposes a novel method named Reasoning Graph-enhanced Exemplar Retrieval (RGER). RGER first quires LLM to generate an initial response, then expresses intermediate problem-solving steps to a graph structure. After that, it employs graph kernel to select exemplars with semantic and structural similarity. Extensive experiments demonstrate the structural relationship is helpful to the alignment of queries and candidate exemplars. The efficacy of RGER on math and logit reasoning tasks showcases its superiority over state-of-the-art retrieval-based approaches. Our code is released at //github.com/Yukang-Lin/RGER.

We consider the computational efficiency of Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to partial differential equations with random coefficients. These arise, for example, in groundwater flow modelling, where a commonly used model for the unknown parameter is a random field. We make use of the circulant embedding procedure for sampling from the aforementioned coefficient. To improve the computational complexity of the MLMC estimator in the case of highly oscillatory random fields, we devise and implement a smoothing technique integrated into the circulant embedding method. This allows to choose the coarsest mesh on the first level of MLMC independently of the correlation length of the covariance function of the random field, leading to considerable savings in computational cost. We illustrate this with numerical experiments, where we see a saving of factor 5-10 in computational cost for accuracies of practical interest.

Blind estimation of intersymbol interference channels based on the Baum-Welch (BW) algorithm, a specific implementation of the expectation-maximization (EM) algorithm for training hidden Markov models, is robust and does not require labeled data. However, it is known for its extensive computation cost, slow convergence, and frequently converges to a local maximum. In this paper, we modified the trellis structure of the BW algorithm by associating the channel parameters with two consecutive states. This modification enables us to reduce the number of required states by half while maintaining the same performance. Moreover, to improve the convergence rate and the estimation performance, we construct a joint turbo-BW-equalization system by exploiting the extrinsic information produced by the turbo decoder to refine the BW-based estimator at each EM iteration. Our experiments demonstrate that the joint system achieves convergence in just 4 EM iterations, which is 8 iterations less than a separate system design for a signal-to-noise ratio (SNR) of 6 dB. Additionally, the joint system provides improved estimation accuracy with a mean square error (MSE) of $10^{-4}$. We also identify scenarios where a joint design is not preferable, especially when the channel is noisy (e.g., SNR=2 dB) and the turbo decoder is unable to provide reliable extrinsic information for a BW-based estimator.

We introduce multiple physics pretraining (MPP), an autoregressive task-agnostic pretraining approach for physical surrogate modeling of spatiotemporal systems with transformers. In MPP, rather than training one model on a specific physical system, we train a backbone model to predict the dynamics of multiple heterogeneous physical systems simultaneously in order to learn features that are broadly useful across systems and facilitate transfer. In order to learn effectively in this setting, we introduce a shared embedding and normalization strategy that projects the fields of multiple systems into a shared embedding space. We validate the efficacy of our approach on both pretraining and downstream tasks over a broad fluid mechanics-oriented benchmark. We show that a single MPP-pretrained transformer is able to match or outperform task-specific baselines on all pretraining sub-tasks without the need for finetuning. For downstream tasks, we demonstrate that finetuning MPP-trained models results in more accurate predictions across multiple time-steps on systems with previously unseen physical components or higher dimensional systems compared to training from scratch or finetuning pretrained video foundation models. We open-source our code and model weights trained at multiple scales for reproducibility.

Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.

This paper proposes a generic method to learn interpretable convolutional filters in a deep convolutional neural network (CNN) for object classification, where each interpretable filter encodes features of a specific object part. Our method does not require additional annotations of object parts or textures for supervision. Instead, we use the same training data as traditional CNNs. Our method automatically assigns each interpretable filter in a high conv-layer with an object part of a certain category during the learning process. Such explicit knowledge representations in conv-layers of CNN help people clarify the logic encoded in the CNN, i.e., answering what patterns the CNN extracts from an input image and uses for prediction. We have tested our method using different benchmark CNNs with various structures to demonstrate the broad applicability of our method. Experiments have shown that our interpretable filters are much more semantically meaningful than traditional filters.

High spectral dimensionality and the shortage of annotations make hyperspectral image (HSI) classification a challenging problem. Recent studies suggest that convolutional neural networks can learn discriminative spatial features, which play a paramount role in HSI interpretation. However, most of these methods ignore the distinctive spectral-spatial characteristic of hyperspectral data. In addition, a large amount of unlabeled data remains an unexploited gold mine for efficient data use. Therefore, we proposed an integration of generative adversarial networks (GANs) and probabilistic graphical models for HSI classification. Specifically, we used a spectral-spatial generator and a discriminator to identify land cover categories of hyperspectral cubes. Moreover, to take advantage of a large amount of unlabeled data, we adopted a conditional random field to refine the preliminary classification results generated by GANs. Experimental results obtained using two commonly studied datasets demonstrate that the proposed framework achieved encouraging classification accuracy using a small number of data for training.