Existing work in scientific machine learning (SciML) has shown that data-driven learning of solution operators can provide a fast approximate alternative to classical numerical partial differential equation (PDE) solvers. Of these, Neural Operators (NOs) have emerged as particularly promising. We observe that several uncertainty quantification (UQ) methods for NOs fail for test inputs that are even moderately out-of-domain (OOD), even when the model approximates the solution well for in-domain tasks. To address this limitation, we show that ensembling several NOs can identify high-error regions and provide good uncertainty estimates that are well-correlated with prediction errors. Based on this, we propose a cost-effective alternative, DiverseNO, that mimics the properties of the ensemble by encouraging diverse predictions from its multiple heads in the last feed-forward layer. We then introduce Operator-ProbConserv, a method that uses these well-calibrated UQ estimates within the ProbConserv framework to update the model. Our empirical results show that Operator-ProbConserv enhances OOD model performance for a variety of challenging PDE problems and satisfies physical constraints such as conservation laws.

Large models training is plagued by the intense compute cost and limited hardware memory. A practical solution is low-precision representation but is troubled by loss in numerical accuracy and unstable training rendering the model less useful. We argue that low-precision floating points can perform well provided the error is properly compensated at the critical locations in the training process. We propose Collage which utilizes multi-component float representation in low-precision to accurately perform operations with numerical errors accounted. To understand the impact of imprecision to training, we propose a simple and novel metric which tracks the lost information during training as well as differentiates various precision strategies. Our method works with commonly used low-precision such as half-precision ($16$-bit floating points) and can be naturally extended to work with even lower precision such as $8$-bit. Experimental results show that pre-training using Collage removes the requirement of using $32$-bit floating-point copies of the model and attains similar/better training performance compared to $(16, 32)$-bit mixed-precision strategy, with up to $3.7\times$ speedup and $\sim 15\%$ to $23\%$ less memory usage in practice.

Integrating and processing information from various sources or modalities are critical for obtaining a comprehensive and accurate perception of the real world in autonomous systems and cyber-physical systems. Drawing inspiration from neuroscience, we develop the Information-Theoretic Hierarchical Perception (ITHP) model, which utilizes the concept of information bottleneck. Different from most traditional fusion models that incorporate all modalities identically in neural networks, our model designates a prime modality and regards the remaining modalities as detectors in the information pathway, serving to distill the flow of information. Our proposed perception model focuses on constructing an effective and compact information flow by achieving a balance between the minimization of mutual information between the latent state and the input modal state, and the maximization of mutual information between the latent states and the remaining modal states. This approach leads to compact latent state representations that retain relevant information while minimizing redundancy, thereby substantially enhancing the performance of multimodal representation learning. Experimental evaluations on the MUStARD, CMU-MOSI, and CMU-MOSEI datasets demonstrate that our model consistently distills crucial information in multimodal learning scenarios, outperforming state-of-the-art benchmarks. Remarkably, on the CMU-MOSI dataset, ITHP surpasses human-level performance in the multimodal sentiment binary classification task across all evaluation metrics (i.e., Binary Accuracy, F1 Score, Mean Absolute Error, and Pearson Correlation).

Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in Fourier/Wavelet space, so the difficulty for solving the coupled PDEs depends on dealing with the coupled mappings between the functions. Towards this end, we propose a coupled multiwavelets neural operator (CMWNO) learning scheme by decoupling the coupled integral kernels during the multiwavelet decomposition and reconstruction procedures in the Wavelet space. The proposed model achieves significantly higher accuracy compared to previous learning-based solvers in solving the coupled PDEs including Gray-Scott (GS) equations and the non-local mean field game (MFG) problem. According to our experimental results, the proposed model exhibits a 2ˆ 4ˆ improvement relative L2 error compared to the best results from the state-of-the-art models. Human perception relies on detecting and processing waves. While our eyes detect waves of electromagnetic radiation, our ears detect waves of compression in the surrounding air.

Integrating and processing information from various sources or modalities are critical for obtaining a comprehensive and accurate perception of the real world. Drawing inspiration from neuroscience, we develop the Information-Theoretic Hierarchical Perception (ITHP) model, which utilizes the concept of information bottleneck. Distinct from most traditional fusion models that aim to incorporate all modalities as input, our model designates the prime modality as input, while the remaining modalities act as detectors in the information pathway. Our proposed perception model focuses on constructing an effective and compact information flow by achieving a balance between the minimization of mutual information between the latent state and the input modal state, and the maximization of mutual information between the latent states and the remaining modal states. This approach leads to compact latent state representations that retain relevant information while minimizing redundancy, thereby substantially enhancing the performance of downstream tasks. Experimental evaluations on both the MUStARD and CMU-MOSI datasets demonstrate that our model consistently distills crucial information in multimodal learning scenarios, outperforming state-of-the-art benchmarks. Remarkably, on the CMU-MOSI dataset, ITHP-DeBERTa surpasses human-level performance in the multimodal sentiment binary classification task across all evaluation metrics (i.e., Binary Accuracy, F1 Score, Mean Absolute Error, and Pearson Correlation).

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively "easy" PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively "hard" PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating conservation constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticities. In each case, it achieves superior predictive performance on downstream tasks.

Ensembling is among the most popular tools in machine learning (ML) due to its effectiveness in minimizing variance and thus improving generalization. Most ensembling methods for black-box base learners fall under the umbrella of "stacked generalization," namely training an ML algorithm that takes the inferences from the base learners as input. While stacking has been widely applied in practice, its theoretical properties are poorly understood. In this paper, we prove a novel result, showing that choosing the best stacked generalization from a (finite or finite-dimensional) family of stacked generalizations based on cross-validated performance does not perform "much worse" than the oracle best. Our result strengthens and significantly extends the results in Van der Laan et al. (2007). Inspired by the theoretical analysis, we further propose a particular family of stacked generalizations in the context of probabilistic forecasting, each one with a different sensitivity for how much the ensemble weights are allowed to vary across items, timestamps in the forecast horizon, and quantiles. Experimental results demonstrate the performance gain of the proposed method.

Chronic obstructive pulmonary disease (COPD) is one of the leading causes of death worldwide, usually associated with smoking and environmental occupational exposures. Prior studies have shown that current COPD diagnosis (i.e., spirometry test) can be unreliable because the test can be difficult to do and depends on an adequate effort from the testee and supervision of the testor. Moreover, the extensive early detection and diagnosis of COPD is challenging. We address the COPD detection problem by constructing two novel COPD physiological signals datasets (4432 medical records from 54 patients in the WestRo COPD dataset and 13824 medical records from 534 patients in the WestRo Porti COPD dataset), demonstrating their complex coupled fractal dynamical characteristics, and performing a rigorous fractional-order dynamics deep learning analysis to diagnose COPD with high accuracy. We find that the fractional-order dynamical modeling can extract distinguishing signatures from the physiological signals across patients with all COPD stages--from stage 0 (healthy) to stage 4 (very severe). We exploit these fractional signatures to develop and train a deep neural network that predicts the suspected patients' COPD stages based on the input features (such as thorax breathing effort, respiratory rate, or oxygen saturation levels). We show that our COPD diagnostics method (fractional dynamic deep learning model) achieves a high prediction accuracy (98.66% 0.45%) on WestRo COPD dataset and can serve as an excellent and robust alternative to traditional spirometry-based medical diagnosis. Our fractional dynamic deep learning model (FDDLM) for COPD diagnosis also presents high prediction accuracy when validated by a dataset with different physiological signals recorded (i.e., 94.01%

Boundary conditions (BCs) are important groups of physics-enforced constraints that are necessary for solutions of Partial Differential Equations (PDEs) to satisfy at specific spatial locations. These constraints carry important physical meaning, and guarantee the existence and the uniqueness of the PDE solution. Current neural-network based approaches that aim to solve PDEs rely only on training data to help the model learn BCs implicitly. There is no guarantee of BC satisfaction by these models during evaluation. In this work, we propose Boundary enforcing Operator Network (BOON) that enables the BC satisfaction of neural operators by making structural changes to the operator kernel. We provide our refinement procedure, and demonstrate the satisfaction of physics-based BCs, e.g. Dirichlet, Neumann, and periodic by the solutions obtained by BOON. Numerical experiments based on multiple PDEs with a wide variety of applications indicate that the proposed approach ensures satisfaction of BCs, and leads to more accurate solutions over the entire domain. The proposed correction method exhibits a (2X-20X) improvement over a given operator model in relative $L^2$ error (0.000084 relative $L^2$ error for Burgers' equation).

Transformer-based models have gained large popularity and demonstrated promising results in long-term time-series forecasting in recent years. In addition to learning attention in time domain, recent works also explore learning attention in frequency domains (e.g., Fourier domain, wavelet domain), given that seasonal patterns can be better captured in these domains. In this work, we seek to understand the relationships between attention models in different time and frequency domains. Theoretically, we show that attention models in different domains are equivalent under linear conditions (i.e., linear kernel to attention scores). Empirically, we analyze how attention models of different domains show different behaviors through various synthetic experiments with seasonality, trend and noise, with emphasis on the role of softmax operation therein. Both these theoretical and empirical analyses motivate us to propose a new method: TDformer (Trend Decomposition Transformer), that first applies seasonal-trend decomposition, and then additively combines an MLP which predicts the trend component with Fourier attention which predicts the seasonal component to obtain the final prediction. Extensive experiments on benchmark time-series forecasting datasets demonstrate that TDformer achieves state-of-the-art performance against existing attention-based models.