Convex optimization method identifies LTV models with smooth or discontinuous dynamics.
problem Identifying LTV dynamical models with smooth or discontinuous time evolution.
method Convex optimization to design cost functions promoting either continuous or discontinuous changes in model coefficients.
result Demonstrates identification of LTV models with either continuous or discontinuous dynamics.
New neural network designs learn contact dynamics efficiently.
problem Learning contact dynamics in robotics from noisy data.
method Physically structured neural networks.
result Data-efficient learning of discontinuous contact events.
New algorithm tackles optimization problems with discontinuous gradients in finance and insurance.
problem Optimization problems with discontinuous stochastic gradients in finance and insurance.
method Langevin dynamics based algorithm e-THεO POULA. result Non-asymptotic error bounds and expected excess risk estimates for e-THεO POULA. This work models overnight rates with jumps and discontinuities, extending classical short-rate models.
problem Capturing the jump behavior and discontinuities in overnight rates for accurate modeling.
method Developed a term structure modeling framework based on overnight rates, accommodating stochastic discontinuities.
result Simple specifications can capture the jump behavior of overnight rates, and explicit valuation formulas are provided.
Let M be a twisted interval bundle over a nonorientable hyperbolizable surface. Let X(M) be the PSL(2,C)-character variety of π_1(M). We examine the dynamics of the action of Out(π_1(M)) on X(M), and in particular, we find an open set on which the action is properly discontinuous that is strictly larger than the interi…
We survey recent work on the dynamics of the outer automorphism group of a word hyperbolic group on spaces of (conjugacy classes of) representations ofthe group into a semi-simple Lie group G. All these results are motivated by the fact that the mapping class group of a closed surface acts properly discontinuously on t…
DGNet solves complex dynamical systems with neural networks and constraints.
problem Real-time accurate solutions for large-scale complex systems.
method Model-constrained discontinuous Galerkin Network (DGNet) for compressible Euler equations.
result DGNet achieves out-of-distribution generalization and improved stability.
This paper analyzes the training dynamics of binary neural networks using information bottleneck.
problem Training binary neural networks is challenging due to discontinuity in activation functions.
method The approach uses the Information Bottleneck principle to analyze BNN training dynamics.
result Training dynamics of BNNs are different from DNNs, with both phases occurring simultaneously.
This study addresses the challenges of dynamic mini-batch sub-sampling in neural network training.
problem Challenges in training neural networks due to dynamic mini-batch sub-sampling.
method Distinguishes between static and dynamic sub-sampling, recasting optimization to find SNN-GPPs.
result SNN-GPPs are less susceptible to sub-sampling-induced discontinuities and better approximate true optima.
New method handles complex systems with discontinuous, heavy-tailed noise.
problem Handling discontinuous, heavy-tailed Lévy noise in stochastic systems.
method Developed nonlocal Kramers-Moyal formulas for SDEs with multiplicative Lévy noise.
result Validated framework for discovering interpretable SDE models from data.
Examines SOFR derivatives pricing and hedging post-LIBOR discontinuation.
problem Pricing and hedging of SOFR derivatives post-LIBOR discontinuation.
method One-factor model based on Vasicek's equation for overnight interest rates dynamics.
result Arbitrage-free pricing and hedging of SOFR derivatives instruments.
Paper introduces a new method to model epidemic dynamics with varying parameters.
problem Capturing discontinuous variations in epidemic model parameters.
method Total variation regularization with Iterated Nelder--Mead optimization.
result The method accurately models epidemic dynamics with instant changes.
New continuous-time optimization algorithms converge in finite time to local minima.
problem Finding local minima in optimization problems.
method Discontinuous dynamical systems with finite-time convergence via Lyapunov-based differential inequality.
result Finite-time convergence to strict local minima with provable settling time.
Decomposes flows with jumps into simpler components.
problem Understanding dynamics of flows with discontinuities.
method Extension of Itô-Ventzel-Kunita formula for stochastic flows with jumps.
result Explicit equations for each component of the decomposition.
Investment and consumption strategies optimized with transaction costs.
problem Optimal investment and consumption strategies with transaction costs.
method Weak dynamic programming and comparison of viscosity solutions.
result Detailed numerical experiments illustrate properties of optimal strategies.
Deep neural network models for efficient uncertainty quantification in multiphase flow.
problem Uncertainty quantification of dynamic multiphase flow in heterogeneous media due to high dimensionality and discontinuities.
method Convolutional encoder-decoder neural network for image-to-image regression, incorporating time as an input.
result Accurate surrogate model capable of characterizing spatio-temporal pressure and saturation fields with limited training data.
Proves nonemptyness of domains for specific group actions.
problem Nonemptyness of domains of proper discontinuity for Anosov groups of affine Lorentzian transformations.
method Proof of nonemptyness of domains of proper discontinuity.
result Proves nonemptyness of domains for Anosov groups of affine Lorentzian transformations.
We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs (N,L), established in [2,3], as a dynamical thickening of the stable…
Neural Laplace models diverse DEs in the Laplace domain for better dynamics.
problem Inadequate ODEs for long-range dependencies and discontinuities.
method Unified framework in Laplace domain, using stereographic map for smoothness.
result Superior performance in diverse DEs, including complex history dependency and abrupt changes.
We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a `small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of disc…
Cut-DeepONet handles discontinuities and sharp transitions in neural operators.
problem Neural operators struggle with discontinuities and sharp transitions in PDEs.
method Two-stage training framework that explicitly models discontinuities via a lifting strategy and input-dependent discontinuity prediction.
result Cut-DeepONet outperforms state-of-the-art methods on benchmark PDEs with low-resolution datasets.
Study proves existence of equilibrium in incomplete economies with discontinuous volatility.
problem Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
method Established existence of solution for Markovian quadratic BSDEs with discontinuous generators using unique continuation and backward uniqueness.
result Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
New domains of discontinuity found for Anosov representations.
problem Understanding Anosov representations acting on homogeneous spaces.
method Constructing open domains of discontinuity for Anosov representations acting on specific homogeneous spaces.
result Describes the largest possible open domains of discontinuity for Zariski dense Anosov representations.
As early as 1972, Penrose - in a purely formal way - introduced a "discontinuous coordinate transformation", which relates a continuous representation of the metric of impulsive pp-waves to a discontinuous one. On the basis of the invertibility concept for generalized functions developed recently by the first author, w…
A math surface has varying perimeter rules.
problem Finding a surface with a non-continuous isoperimetric profile.
method Constructed a 2D Riemannian manifold.
result Discovered a surface with a discontinuous isoperimetric profile.
We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous- semimartingale. A uniqueness theorem is given for the solutions of this equation. This result generalizes Dupire…
GOALS improves learning rate selection for dynamic MBSS in deep learning.
problem Challenges in selecting learning rates for dynamic MBSS in deep learning.
method Gradient-only approximation line search (GOALS) for dynamic MBSS loss functions.
result GOALS reduces model errors in multimodal cases.
Study the discontinuity of functions not embeddable in Euclidean space.
problem Understanding discontinuity of non-embeddable functions.
method Define a modulus of discontinuity and establish lower bounds.
result Quantified nonembeddability results and topological Tverberg theorem.
This note introduces and studies an open set of PSL(2,C) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that th…
Characterizes discontinuous motions of limit sets in quasi-Fuchsian groups.
problem Understanding discontinuous motions of limit sets in quasi-Fuchsian groups.
method Characterization of discontinuous motions through Cannon-Thurston maps for algebraically convergent groups.
result Completely characterizes discontinuous motions of limit sets.
Bayesian nonparametric discontinuity design improves causal inference without randomization.
problem Causal inference in non-randomized settings with additional assumptions.
method Bayesian model comparison and Gaussian process regression.
result BNDD can detect discontinuities of any order and spectral features.
Study new symmetries in non-symmetric spaces and discontinuous groups.
problem Analyze symmetries in non-symmetric homogeneous spaces and discontinuous groups.
method Investigate discrete series, discontinuous groups, and analysis on pseudo-Riemannian spaces.
result New insights into symmetries of non-symmetric homogeneous spaces and discontinuous groups.
Improved method for estimating derivatives of discontinuous functions using stochastic algorithmic differentiation and regression.
problem High Monte-Carlo error in finite difference approximation of discontinuous functions.
method Combining stochastic algorithmic differentiation and regression to estimate derivative of expectations of discontinuous functions.
result Reduction in Monte-Carlo error through decoupling integration of Dirac delta and conditional expectation.
Paper finds surface groups can deform in reductive symmetric spaces.
problem Finding deformations of discontinuous groups in reductive symmetric spaces.
method Analyzing Zariski dense surface subgroups and their deformations.
result Surface groups of high genus can deform in reductive symmetric spaces.
A new method for unsupervised disentanglement using axis-aligned cliffs.
problem Unsupervised disentanglement of latent factors under nonlinear maps.
method Encouraging axis-aligned discontinuities (cliffs) in the estimated density of factors.
result Cliff method outperforms baselines on disentanglement benchmarks.
Donors who defer their donations volunteer less in the future.
problem Volunteer labor can be less beneficial to charities than its costs.
method Regression discontinuity design with a procedure to handle manipulation.
result Donor manipulation invalidates standard regression discontinuity design, but a new method provides partial identification bounds.
Develops a framework for modeling interest rate markets with jumps.
problem Stochastic discontinuities in interest rate markets.
method Extended HJM framework with stochastic discontinuities, affine semimartingales.
result Fundamental theorem of asset pricing based on NAFLVR.
New method estimates active subspaces for jump-discontinuous functions.
problem Estimating active subspaces for discontinuous functions like ABMs.
method Extending active subspaces to discontinuous functions, using Gaussian process.
result Identifies important parameters in ABM simulations of refugee movement.
This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing the current knowledge of possible global forms of pseudo-Riemannian manifolds with constant curvatures, and discuss what kind of …
Study of SU(2,1) character varieties on one-holed torus.
problem Characterize representations of mapping class group on SU(2,1) character variety.
method Explicit description of SU(2,1) character variety, use of Farey graph adaptation, and mapping class group action analysis.
result Description of an open domain of discontinuity for mapping class group action.
The first known example of a complete Riemannian manifold whose isoperimetric profile is discontinuous is given.
Paper analyzes error in stochastic approximation for discontinuous functions.
problem Estimating expected error in discontinuous stochastic approximation.
method Uses finite differences and O(n−1/5) error estimate for discontinuous functions. result Achieves error estimate of O(n−1/5) for discontinuous stochastic representation. Bayesian emulator tackles models with discontinuities efficiently.
problem Uncertainty quantification of models with multiple discontinuities.
method TENSE framework, designed correlation structures, single emulator object.
result Efficient emulation of complex models with multiple discontinuities.
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.
We study the properly discontinuous and isometric actions on the unit sphere of infinite dimensional Hilbert spaces and we get some new examples of Hilbert manifold with costant positive sectional curvature. We prove some necessary conditions for a group to act isometrically and properly discontinuously and in the case…
We construct continuous-time equilibrium models based on a finite number of exponential utility investors. The investors' income rates as well as the stock's dividend rate are governed by discontinuous Levy processes. Our main result provides the equilibrium (i.e., bond and stock price dynamics) in closed-form. As an a…
The study explores deformations of discrete subgroups in non-compact homogeneous spaces.
problem Addressing the proper discontinuity of discrete subgroups in non-compact homogeneous spaces.
method Classification results for deformations of standard discontinuous groups in pseudo-Riemannian homogeneous spaces.
result Conditions for local rigidity and Zariski-dense deformations in standard quotients.
Paper analyzes SGHMC for non-convex optimization with discontinuous gradients.
problem Training neural networks with ReLU activation.
method Non-asymptotic convergence analysis of SGHMC with discontinuous gradients.
result Explicit upper bounds for expected excess risk in non-convex optimization.