We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…
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48 results for “domain decomposition”
New risk decompositions clarify domain adaptation issues.
problem Domain adaptation challenges with different training and test distributions.
method Representation Bayesian Risk Decompositions, hybrid argument.
result Clarifies factors (2) and (3) as reasons for generalization failure.
APINNs improve physics-informed neural networks through flexible domain decomposition.
problem Improving physics-informed neural networks (PINNs) for solving partial differential equations (PDEs).
method Introduces a trainable gate network for soft domain decomposition, allowing flexible parameter sharing and improved generalization.
result APINNs significantly improve PINNs and XPINNs, demonstrating better performance on various types of PDEs.
New method solves elliptic equations on manifolds without grids.
problem Solving elliptic equations on complex manifolds.
method Numerical domain decomposition method avoiding global grids.
result Method validated on specific 4D manifolds.
CSD learns a common component for domain generalization, outperforming existing methods.
problem Training models to generalize across unseen domains.
method CSD decomposes the model into a common and specific component, discarding the latter.
result CSD outperforms state-of-the-art domain generalization methods.
New DDMs use neural networks for solving equations on manifold shapes.
problem Solving equations on complex, high-dimensional shapes.
method Physics-informed neural networks combined with domain decomposition methods.
result Validated methods work well on various shapes in high dimensions.
This work introduces a bias-variance decomposition for proper scores, improving uncertainty estimation in predictive models.
problem Reliable uncertainty estimation for predictions in safety-critical applications, especially under domain drift.
method Developed a general bias-variance decomposition for proper scores, introducing the Bregman Information as the variance term.
result The decomposition provides novel formulations for different predictive tasks, including classification and model ensembles.
Analyzes vector fields in polytope decompositions, proving curve finiteness.
problem Analyzing vector fields in polytope decompositions.
method Proves integral curves are chopped into finitely many pieces by polytope decompositions.
result Finiteness of edge flips in discrete Yamabe flow.
New method for decomposing high-dimensional parametric domains using PCA and inverse projection.
problem Decomposing high-dimensional parametric domains efficiently.
method Iterative Principal Component Analysis (PCA) and inverse projection methods.
result The proposed method effectively reconstructs the original domain from lower-dimensional data.
KEDformer improves long-term time series forecasting with seasonal-trend decomposition.
problem Accurate long-term predictions in energy, finance, and meteorology.
method Knowledge extraction-driven framework integrating seasonal-trend decomposition.
result KEDformer enhances model's ability to capture short-term and long-term patterns.
This paper constructs a continuous decomposition of the Sierpiński curve into acyclic continua one of which is an arc. This decomposition is then used to construct another continuous decomposition of the Sierpiński curve. The resulting decomposition space is homeomorphic to the continuum obtained from taking the Sierpi…
New method explains ML performance gaps without causal knowledge.
problem Understanding why ML algorithms perform differently across domains.
method Nonparametric hierarchical decomposition framework.
result Detailed variable-level explanations for performance gaps.
In this paper, we present a novel framework incorporating a combination of sparse models in different domains. We posit the observed data as generated from a linear combination of a sparse Gaussian Markov model (with a sparse precision matrix) and a sparse Gaussian independence model (with a sparse covariance matrix). …
New method evaluates multiple social disparities using machine learning.
problem Reduction of educational disparities across multiple dimensions.
method Triply-Robust Machine Learning Approach for Causal Decomposition Analysis.
result Simultaneous interventions across multiple domains reduce disparities.
The paper develops axioms for uniquely decomposing functions with real arguments.
problem Decomposing functions with real arguments while preserving their overall structure.
method Developing axioms to uniquely decompose Borel measurable functions.
result Unique decompositions for all Borel measurable functions are achieved.
We compute the Dolbeault cohomology of geodesically convex domains contained in Cousin groups which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of Oeljeklaus-Toma manifolds and in particular the fact that the Hodge decomposition holds for their cohomol…
EQD model improves domain-specific QA by 0.6% to 10.5%.
problem Challenges in domain-specific quantitative reasoning for LLMs.
method Two-step fine-tuning framework guided by a reward function.
result EQD outperforms state-of-the-art models and prompting strategies.
FreDN: Spectral Disentanglement for Time Series Forecasting via Learnable Frequency Decompositionstat.ML
FreDN separates trends and periodicities in non-stationary time series forecasts.
problem Spectral entanglement and computational burden in frequency-domain methods for non-stationary time series.
method FreDN introduces a learnable Frequency Disentangler module to separate trend and periodic components directly in the frequency domain, and uses a ReIm Block to reduce complexity.
result FreDN outperforms state-of-the-art methods by up to 10% on long-term forecasting benchmarks.
A new method tackles Bayesian inverse problems with complex PDEs.
problem Bayesian inverse problems with expensive forward model evaluations and high-dimensional priors.
method Domain-decomposed variational auto-encoder Markov chain Monte Carlo (DD-VAE-MCMC) method.
result The method efficiently solves Bayesian inverse problems in parallel and low-dimensional latent spaces.
Dual energy computed tomography (DECT) imaging plays an important role in advanced imaging applications due to its material decomposition capability. Image-domain decomposition operates directly on CT images using linear matrix inversion, but the decomposed material images can be severely degraded by noise and artifact…
Paper tackles domain generalization by minimizing domain-based covariance.
problem Training data and test data have different distributions, leading to poor generalization.
method Find a central subspace minimizing domain-based covariance while preserving functional relationships.
result The proposed method achieves better generalization performance on unseen test datasets.
Researchers prove hot spots conjecture for Gaussian spaces.
problem Hot spots conjecture for Gaussian domains.
method Variational principle for Hodge Laplacian on weighted manifolds and Hodge decomposition.
result First nontrivial eigenfunction extrema are on the boundary for specified domains.
We show that every smooth closed oriented four-manifold admits a decomposition into two co- dimension zero submanifolds with common boundary. Each of these submanifolds carries a structure of a symplectic manifold with pseudo-convex boundary. This imply, in particular, that every smooth closed simply-connected four-man…
We show the existence of a thick thin decomposition of the domain of a pseudo holomorphic curve with boundary. The geometry of the thick part is bounded uniformly in the energy. Furthermore, in the thick part, there is a uniform bound on the differential which is exponential in the energy. The thin part consists of ann…
Pattern Decomposition with Complex Combinatorial Constraints: Application to Materials Discoverycs.AI
Identifying important components or factors in large amounts of noisy data is a key problem in machine learning and data mining. Motivated by a pattern decomposition problem in materials discovery, aimed at discovering new materials for renewable energy, e.g. for fuel and solar cells, we introduce CombiFD, a framework …
DecompKAN improves time series forecasting accuracy and transparency.
problem Accurate and transparent time series forecasting in scientific domains.
method Combines decomposition, patching, normalization, and B-spline KAN edge functions.
result Achieves best or tied-best MSE on 20 of 36 comparisons across 9 datasets.
Improves neural network verification by merging abstract domains and Lagrangian methods.
problem Prove provable bounds for neural network outputs given input ranges.
method Uses zonotopes within a Lagrangian decomposition to verify deep neural networks.
result Yields bounds that improve upon existing techniques in both time and tightness.
We introduce an architecture based on deep hierarchical decompositions to learn effective representations of large graphs. Our framework extends classic R-decompositions used in kernel methods, enabling nested part-of-part relations. Unlike recursive neural networks, which unroll a template on input graphs directly, we…
XPINNs improve generalization by decomposing PDEs but may overfit.
problem Understanding when XPINNs outperform PINNs in generalization.
method Theoretical bounds and empirical validation.
result XPINNs improve generalization by decomposing complex PDEs but may overfit.
Paper introduces a new method for efficient portfolio risk quantification.
problem Efficiently quantify risk in large portfolios with many trades and few dominant risk factors.
method Combines Fourier-cosine series with tensor decomposition techniques for dimension reduction.
result Achieves relative errors below 0.1% with significant runtime improvement.
Adaptive tensor modeling preserves continuity in multidimensional data.
problem Discretization of continuous multidimensional data loses important information.
method Functional Tucker decomposition (FTD) with RKHS modeling.
result FTD enables adaptive and expressive tensor modeling.
Many advanced Learning from Demonstration (LfD) methods consider the decomposition of complex, real-world tasks into simpler sub-tasks. By reusing the corresponding sub-policies within and between tasks, they provide training data for each policy from different high-level tasks and compose them to perform novel ones. E…
Generalizes PCA and ICA for continuous-time signals using neural networks.
problem Low-rank decomposition of continuous-time vector-valued signals.
method Implicit neural network framework to learn numerical approximations of PCA and ICA.
result Unified approach to PCA and ICA in continuous domain, enforcing decorrelation and independence.
Gaussian processes classify graphs using vertex and edge features.
problem Graph classification in machine learning.
method Transform graph features into spectral Euclidean features, apply Hodge decomposition.
result Gaussian processes can classify graphs using vertex and edge features.
The Geometry of Filteringmath.DG
Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …
TRIM improves interpretability of deep neural networks in cosmology.
problem Understanding which features a deep neural network uses in a transformed space.
method TRIM (Transformation IMportance) attributes importances to features in a transformed space.
result Combining TRIM with contextual decomposition helps identify physical features learned by DNNs.
The Blaschke rolling disk theorem is extended to non-convex domains.
problem Classical inclusion principle for non-convex domains.
method Geometric conditions based on curvature, algorithm for decomposition.
result Necessary and sufficient conditions for rolling disks in non-convex domains.
In this paper, we consider domain-invariant deep learning by explicitly modeling domain shifts with only a small amount of domain-specific parameters in a Convolutional Neural Network (CNN). By exploiting the observation that a convolutional filter can be well approximated as a linear combination of a small set of dict…
In recent work, we have proven uniform decay bounds for solutions of the wave equation on a Schwarzschild exterior, in particular, the uniform pointwise estimate , which holds throughout the domain of outer communications, where is an advanced Eddington-Finkelstein coordinate, $v_+=\ma…
Paper introduces a PDE-free method for decomposing forces in any dimension.
problem Analyzing non-conservative forces in arbitrary dimensions.
method Geometric decomposition using homotopy operator and Frobenius theorem.
result Decomposes forces into gradient and antiexact components, characterizing curl forces.
We present a novel method for learning a set of disentangled reward functions that sum to the original environment reward and are constrained to be independently obtainable. We define independent obtainability in terms of value functions with respect to obtaining one learned reward while pursuing another learned reward…
Across numerous applications, forecasting relies on numerical solvers for partial differential equations (PDEs). Although the use of deep-learning techniques has been proposed, actual applications have been restricted by the fact the training data are obtained using traditional PDE solvers. Thereby, the uses of deep-le…
Dynamic Mode Decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of non-linear systems from experimental datasets. Recently, several attempts have extended DMD to the context of low-rank approximations. This extension is of particular interest for reduced-order modeling in various applicative …
Revisits orbital minimization for neural operator decomposition.
problem Training neural networks to approximate eigenfunctions of operators.
method Adapts orbital minimization method (OMM) for neural networks.
result Justifies broader applicability of OMM in modern learning pipelines.
VDA improves disentanglement of latent representations in complex signals.
problem Learning disentangled and interpretable representations in nonstationary, high-dimensional time-evolving signals.
method Variational decomposition autoencoding (VDA) framework, incorporating signal decomposition, contrastive self-supervised task, and variational prior approximation.
result DecVAEs surpass state-of-the-art VAE-based methods in disentanglement quality and generalization.
Enhances Gaussian processes with spherical features for better scalability and flexibility.
problem Lack of representation learning in Gaussian processes compared to deep neural networks.
method Introduces spherical inter-domain features to improve GP approximation and scalability.
result The method alleviates limitations and improves scalability compared to alternative strategies.
Paper proves optimal decomposition for matrix fields, reducing convex integration steps.
problem Optimizing decomposition of symmetric matrix fields for convex integration.
method Algebraic geometry and topology applications to prove optimality.
result Optimal decomposition with fewer rank-one terms, improving Hölder regularity.
Optimal transport aligns rotated linear regression models across domains.
problem Aligning rotated linear regression models across domains with differing statistical properties.
method Combines K-means clustering, OT, and SVD to estimate rotation angle and adapt regression model.
result Optimal transport map recovers underlying rotation in .