Introduce Collapsed Effective Operators for higher-order structures.
problem Existing spectral operators decompose topology into separate ranks, leaving practitioners to fuse information back to vertices.
method Introduce Collapsed Effective Operators via Schur complementation of a graded Laplacian.
result Preserves positive semi-definiteness, lowers system energy under higher-order connectivity.
New method identifies latent treatment effects from proxy models.
problem Identifying heterogeneous treatment effects under unobserved confounding.
method Compressed observable operator and spectral analysis of treatment effects.
result Eigenvalues of the operator represent latent treatment effects.
Physics-informed WNO learns PDE solutions without labeled data.
problem Data-hungry nature of WNO framework.
method Physics-informed WNO for learning PDE solutions.
result Validated and illustrated with four nonlinear systems.
Study Laplace operators in adiabatic limit of fibre bundles.
problem Understanding Laplace-type operators in the adiabatic limit of complex vector bundles.
method Analyse the adiabatic limit of fibre bundles with compact fibres, proving existence of effective operators providing asymptotics.
result Existence of effective operators providing asymptotics to any order in ε for Laplace-type operators H on an almost-invariant subspace of L^2(E).
Analyzes tunneling effects for Schrödinger operators on vector bundles.
problem Tunneling effects in quantum systems with multiple potential wells.
method Quasimodes and WKB analysis near potential wells, interaction matrix for coupling between wells.
result Polynomial prefactor for exponentially small eigenvalue splitting determined by dimension of minimal geodesics.
New neural operators model turbulence with memory and randomness.
problem Modeling turbulence in complex fluid dynamics with memory and randomness.
method Symmetrized activation functions, fractional derivatives, and stochastic noise.
result Theoretical guarantees for approximation quality in turbulent phenomena.
dynoNet learns dynamical systems using linear operators.
problem Learning complex dynamical systems.
method dynoNet uses linear dynamical operators for sequence modeling and system identification.
result dynoNet effectively identifies systems on benchmarks.
Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
New reinforcement learning operators improve performance and robustness.
problem Improving reinforcement learning algorithms to handle approximation errors.
method Developed a new family of robust stochastic operators.
result Preserves optimality and increases action gap on sample paths.
ISOKANN learns collective variables and effective dynamics for metastable transitions.
problem Understanding metastable transitions in complex molecular systems.
method Integrates Koopman operators with neural networks to extract CVs and effective dynamics.
result Reconstructs coarse-grained kinetics and reproduces transition times across barriers.
Paper uses Koopman operator and Nyström method for efficient nonlinear control.
problem Control of nonlinear dynamical systems.
method Combines Koopman operator framework with Nyström approximation for kernel methods.
result Theoretical guarantees on the convergence rates of the approximated Riccati operator and regulator objective.
Improved DeepONets for PDE solution operators with adaptive re-weighting and new architecture.
problem Training DeepONets for PDE solution operators without paired data.
method Adaptive re-weighting of training examples and novel network architecture.
result Consistently improved predictive accuracy by a factor of 10-50x.
Introduces a new spectral geometry framework with dissipative data.
problem Deforming spectral triples with dissipative Lindblad operators.
method Lindblad-deformed spectral geometry framework with heat-kernel asymptotics.
result First nontrivial dissipative effect appears at order gamma^4.
We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eige…
Bayesian inference using particle flow and neural operators.
problem Efficiently updating beliefs with new data.
method ODE-based neural operator for particle flow, meta-learning for generalization.
result Generalization across different priors, observations, and sequential inference.
Effective quantum dynamics on a thin Möbius strip approximated by a flat model.
problem Quantum dynamics on a Möbius strip with zero width.
method Norm-resolvent convergence to an unconventional flat model with explicit spectrum.
result Spectral properties of the curved Möbius strip are well approximated by a flat model.
Physics-informed neural networks improve by measuring effective dimensionality of constraints.
problem Task interference in physics-informed neural networks due to shared parameter space.
method Introduce effective dimensionality (deff) as an operator invariant to quantify constraints. result Effective dimensionality measures unconstrained parameter directions, independent of network architecture.
We study a regular closure operator in the category of quandles. We show that the regular closure operator and the pullback closure operator corresponding to the reflector from the category of quandles to its full subcategory of trivial quandles coincide, we give a simple description of this closure operator, and analy…
Geometrically computes superpotentials for certain 4D N=2 theories.
problem Computing effective twisted superpotentials for 4D N=2 theories.
method Spectral networks and abelianization to compute generating functions of brane opers.
result Geometric recipe for computing effective twisted superpotentials.
Framework learns to optimize tensor programs for various hardware.
problem Manual optimization of tensor operators for deep learning limits applicability and increases engineering costs.
method Learning-based statistical cost models guide tensor operator implementations over billions of variants.
result Framework delivers performance competitive with hand-tuned libraries across multiple hardware targets.
Paper learns Koopman operator from sparse data, escaping function space constraints.
problem Learning Koopman operator from non-closed function spaces.
method Operator stochastic approximation algorithm using conditional mean embeddings (CME).
result Online sparse learning algorithm with trajectory-based sampling guarantees.
Functional determinant for mixed signature sphere products depends on sphere dimensions and parity.
problem Determining the functional determinant for scalar fields on mixed signature sphere products.
method Analyzing the GJMS operator on SqimesSp to derive the functional determinant. result The functional determinant depends only on the total dimension and parity of the sphere dimensions.
SCOPE-FE improves feature engineering efficiency for high-dimensional datasets.
problem Expanding and reducing feature space in tabular learning becomes computationally expensive with increased dimensionality.
method SCOPE-FE controls the search space by regulating operator and feature-pair spaces, using OperatorProbing and FeatureClustering.
result SCOPE-FE reduces feature engineering time while maintaining competitive predictive performance.
Paper proposes adaptive parameter selection for KGD algorithms.
problem Improving parameter selection for kernel-based gradient descent.
method Integrates bias-variance analysis with splitting method, introduces empirical effective dimension.
result Adaptive parameter selection strategy achieves optimal generalization error bound.
We consider differential operators on a supermanifold of dimension 1∣1. We define non-degenerate operators as those with an invertible top coefficient in the expansion in the "superderivative" D (which is the square root of the shift generator, the partial derivative in an even variable, with the help of an odd ind…
CViT learns complex physical systems using vision transformer techniques.
problem Learning maps between infinite-dimensional function spaces in scientific machine learning.
method Combines vision transformer encoder, grid-based coordinate embedding, and cross-attention mechanism.
result Achieves state-of-the-art performance on multiple benchmarks, often surpassing larger models.
Bayesian approach learns linear operators from noisy data.
problem Learning linear operators from noisy data in infinite-dimensional spaces.
method Bayesian approach with Gaussian priors.
result Establishes posterior contraction rates and generalization error guarantees.
Paper estimates upper bound of analytic torsion under Arakelov metric.
problem Estimating the analytic torsion under Arakelov metric.
method Defined and analyzed the regularized determinant of Laplacian, providing an asymptotic upper bound.
result The logarithm of analytic torsion is asymptotically upper bounded by g for g>1. New lattice Dirac operator index method for curved boundaries.
problem Defining Dirac operator indices for curved boundaries and gravitational backgrounds.
method Employing K-theory and spectral flow to classify Wilson Dirac operator.
result Mod-2 index defined in both even and odd dimensions.
Extracts compact non-Markovian closure models from data.
problem Deriving reduced order representations of dynamical systems with memory effects.
method Operator inference using sparse polynomial regression and neural networks.
result The extracted models are compact and non-Markovian, capturing underlying dynamics.
New method speeds up Bayesian inverse problem solving with neural operators.
problem Solving infinite-dimensional Bayesian inverse problems with high computational cost.
method Delayed-acceptance geometric MCMC driven by derivative-informed neural operator surrogates.
result Significant speedup in generating posterior samples (3-9 times faster).
Develops a framework for learning nonlinear operators using Mercer kernels.
problem Learning nonlinear operators between infinite-dimensional spaces.
method Stochastic approximation framework with Mercer operator-valued kernels.
result Establishes dimension-free polynomial convergence rates for nonlinear operator learning.
Extends expected value framework for cost-sensitive causal decision-making.
problem Optimizing operational decision-making with cost-sensitive causal classification.
method Introduces a cost-sensitive decision boundary based on estimated individual treatment effects, positive outcome probability, and cost parameters.
result Effective in maximizing expected causal profit, outperforming cost-insensitive ranking approach.
New kernels allow learning from non-separable data.
problem Learning from non-separable data.
method Introducing entangled kernels and a two-step algorithm.
result Efficient algorithm for learning entangled kernels.
Agent learns optimal control inputs for plants with unknown parameters.
problem Optimal control for systems with unknown and changing parameters.
method Personalized control inputs based on stochastic dynamics.
result Demonstrated effectiveness on simulated system.
CD estimates demand by separating promotions from a stable base, reducing inventory costs.
problem Operational volatility in demand forecasts leads to excessive safety stock and inventory costs.
method Contextual Deconvolution (CD) decomposes demand into a smooth base and promotion-driven shocks.
result CD reduces inventory costs by stabilizing demand forecasts, but under-provisions event spikes.
Proposes a new method to learn operators for stochastic problems using DeepONet with autoencoder.
problem Efficiently solve forward and inverse stochastic problems with limited data.
method MultiAuto-DeepONet, a multi-resolution autoencoder DeepONet model.
result The model effectively handles high-dimensional stochastic inputs and reduces the number of trainable parameters.
We generalise the analysis in [arXiv:0904.1744] to superspace, and explicitly prove that for any embedding of surface operators in a general, twisted N=2 pure abelian theory on an arbitrary four-manifold, the parameters transform naturally under the SL(2,Z) duality of the theory. However, for nontrivially-embedded surf…
Study satellite operations on knot invariant θ, proving additivity and distinguishing knots.
problem Behavior of knot invariant θ under satellite operations.
method Proved additivity, introduced computational tool, verified conjecture for 2977 knots.
result Distinguished knots using invariant θ and verified conjecture for 2977 primes.
Researchers derive asymptotic expansions for thermoelastic operators on manifolds.
problem Determining precise geometric information from thermoelastic spectra.
method Asymptotic expansions with Dirichlet and Neumann boundary conditions.
result Explicit calculation of first two coefficients for volumes.
PILNO uses neural operators to solve PDEs efficiently on point clouds.
problem Solving partial differential equations (PDEs) on point cloud data efficiently.
method Physics-informed low-rank neural operator framework combining low-rank kernel approximations and an encoder-decoder architecture.
result PILNO efficiently approximates solution operators of PDEs on point cloud data, satisfying PDE constraints and boundary conditions.
GATES improves neural architecture search by modeling operations as information transformation.
problem Improving predictor-based neural architecture search efficiency.
method GATES models operations as information transformation, covering both node and edge cell search spaces.
result GATES boosts sample efficiency and improves predictor performance.
We find flat band Hamiltonians and Ginsparg-Wilson relations for symmetry classes.
problem Finding flat band Hamiltonians and Ginsparg-Wilson relations for symmetry classes.
method Integrating out the additional bulk direction to obtain effective Dirac operators and then deriving flat and overlap Dirac operators.
result Established Ginsparg-Wilson relations and mod-two index theorems for each symmetry class.
Formula proves Euler characteristic of singularized surfaces.
problem Calculating the Euler characteristic of singularized surfaces.
method Three operations: collapsing, zipping, and double loop identification.
result Formula for Euler characteristic of singularized surfaces.
Statistical depth metrics help identify risky power grid scenarios.
problem Identifying extreme scenarios for risk mitigation in power grid planning.
method Functional depth metrics for sub-selecting outlying scenarios.
result The proposed approach effectively identifies risky scenarios for operational risk mitigation.
New method shows Hessian estimator from random samples converges to true Hessian on complex manifolds.
problem Uncertainty in Hessian estimator accuracy on complex manifolds with boundaries and nonuniform sampling.
method Locally fitting quadratic polynomials, rigorous theoretical analysis under mild conditions.
result The Hessian estimator asymptotically converges to the true Hessian, even near boundaries.
A new RL algorithm POWR learns world models to estimate action-values.
problem Inaccessibility of explicit action-value functions in RL.
method Learning a world model using conditional mean embeddings and deriving action-value function via matrix operations.
result POWR algorithm converges to global optimum with proven rates.
Softmax Bellman operator improves Q-function performance in RL despite sub-optimality.
problem Softmax Bellman operator's impact on value functions in RL is problematic.
method Revisited theoretical properties of softmax Bellman operator, proving convergence and overestimation reduction.
result Softmax Bellman operator leads to superior policies in practice, even outperforming double Q-learning.