The paper compares PINN methods for solving drift-diffusion equations on metric graphs.
problem Solving drift-diffusion equations on metric graphs using machine learning.
method Comparison of physics-informed neural networks (PINNs) for solving drift-diffusion equations on metric graphs.
result PINNs offer a flexible and versatile tool for solving parameter identification or optimization problems on metric graphs.
New method identifies SDE drift and diffusion from temporal data.
problem Learning SDE parameters from temporal data, especially in noisy or incomplete data.
method Entropy-regularized optimal transport, APPEX algorithm.
result Can almost always recover drift and diffusion from temporal marginals.
We derive expressions for the first three moments of the decision time (DT) distribution produced via first threshold crossings by sample paths of a drift-diffusion equation. The "pure" and "extended" diffusion processes are widely used to model two-alternative forced choice decisions, and, while simple formulae for ac…
Motivated by a probabilistic approach to Kahler-Einstein metrics we consider a general non-equilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasi-convex N-particle interaction energy. We show that a deterministic "macroscopic" evolution…
The paper enhances preference learning by incorporating response time data.
problem Lack of temporal information in user decision-making for reward model learning.
method Integrates response time alongside binary choice data using the EZ model and Neyman-orthogonal loss functions.
result Response time-augmented approach reduces error rates from exponential to polynomial scaling, improving sample efficiency.
In quantitative finance, we often model asset prices as semimartingales, with drift, diffusion and jump components. The jump activity index measures the strength of the jumps at high frequencies, and is of interest both in model selection and fitting, and in volatility estimation. In this paper, we give a novel estimat…
Response time improves alignment with diverse human preferences.
problem Standard aggregation of feedback ignores heterogeneity and anonymity.
method Augmenting feedback with response time data and modeling decisions with DDM.
result Estimator of heterogeneous preferences converges to true average preference.
Study reveals attention mechanism's similarity computation parallels traditional machine learning.
problem Understanding the essence and principles of attention mechanism in deep learning.
method Examined classic metrics and vector space properties in manifold learning, clustering, and supervised learning to identify key characteristics of similarity computation and information propagation.
result Self-attention mechanism in deep learning adheres to the same principles but operates more flexibly and adaptively.
A new SNN model explains decision-making with learning and spiking neurons.
problem Lack of learning mechanism in existing models for decision-making.
method Proposes a Spiking Neural Network (SNN) model that incorporates a learning mechanism and uses multivariate Hawkes processes.
result Shows a coupling between DDM and Poisson counter models and derives a DDM from a Hawkes network of spiking neurons.
We study long-term growth-optimal strategies on a simple market with linear proportional transaction costs. We show that several problems of this sort can be solved in closed form, and explicit the non-analytic dependance of optimal strategies and expected frictional losses of the friction parameter. We present one der…
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma z calculus and z--Bochner's formula. result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
This note is concerned with accurate and computationally efficient approximations of moments of Gaussian random variables passed through sigmoid or softmax mappings. These approximations are semi-analytical (i.e. they involve the numerical adjustment of parametric forms) and highly accurate (they yield 5% error at most…
Efficiently reconstructs jump-diffusion processes from data using neural networks.
problem Reconstructing jump-diffusion processes from data.
method Temporally decoupled squared Wasserstein distance method using parameterized neural networks.
result Enhanced reconstruction of jump-diffusion processes from data.
Kernel method approximates dynamical operators from data.
problem Estimating eigenfunctions of dynamical operators from data.
method Kernel-based approach in reproducing kernel Hilbert spaces.
result Eigenfunctions estimated via matrix eigenvalue problems.
The proposed model modifies option pricing formulas for the basic case of log-normal probability distribution providing correspondence to formulated criteria of efficiency and completeness. The model is self-calibrating by historic volatility data; it maintains the constant expected value at maturity of the hedged inst…
Geometric formalism views optimization algorithms as discrete connections, revealing their algebraic curvature and flatness properties.
problem Understanding and optimizing the behavior of iterative optimization algorithms.
method Introducing a geometric and operator-theoretic formalism where optimization algorithms are encoded by coupled channels (drift and diffusion) whose algebraic curvature measures the deviation from ideal reversibility.
result Flat connections correspond to methods whose updates commute up to higher order, achieving minimal numerical dissipation and preserving stability.
Study infinite-depth limits of neural networks with fixed width.
problem Understanding the behavior of neural networks as depth increases with fixed width.
method Analyzing finite-width residual networks with random Gaussian weights, focusing on the infinite-depth limit.
result The pre-activations converge to a zero-drift diffusion process, differing from the infinite-width limit.
Proposes a method to estimate SDE noise from a single trajectory.
problem Estimating SDE noise from a single data trajectory without ergodicity or stationarity.
method Combining Taylor expansions, Girsanov transformations, and drift function's initial value for drift and noise estimation.
result First SSISDE algorithm capable of identifying SDE dynamics from a single trajectory.
An open question in systems and computational neuroscience is how neural circuits accumulate evidence towards a decision. Fitting models of decision-making theory to neural activity helps answer this question, but current approaches limit the number of these models that we can fit to neural data. Here we propose a unif…
In this paper we introduce kinetic equations for the evolution of the probability distribution of two goods among a huge population of agents. The leading idea is to describe the trading of these goods by means of some fundamental rules in price theory, in particular by using Cobb-Douglas utility functions for the bina…
We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential Lévy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent Lévy measure. Using techniques from regular perturba…
New method infers population dynamics from snapshots using path space optimization.
problem Recover dynamics of a population from its temporal marginals.
method Grid-free algorithm using Schrödinger bridges coupled via noisy gradient descent in mean-field limit.
result Global convergence to min-entropy estimator with end-to-end theoretical guarantees.
Study improves robustness of Bayesian inference for cognitive models.
problem Outliers and contaminants affect parameter estimation in cognitive models.
method Robustness of parameter estimation using amortized Bayesian inference (ABI) with neural networks.
result Introducing contaminants from a Cauchy distribution increases robustness.
Develops a new method to discover stochastic systems with non-Gaussian noise.
problem Discovering governing laws from complex systems with non-Gaussian noise.
method Theoretical framework and numerical algorithm to extract stochastic differential equations with Gaussian and non-Gaussian noise.
result Demonstrated the efficacy and accuracy of the approach on various systems.
We develop a variational framework for SDEs driven by fractional noise.
problem Capturing long-term dependencies in SDEs driven by fractional noise.
method Markov approximation of fractional Brownian motion, variational inference, neural networks.
result Efficient variational inference of posterior path measures for neural-SDEs.
Neural Lévy model improves risk and density forecasting for financial returns.
problem Financial returns exhibit heavy tails, volatility clustering, and jumps.
method Proposes a neural Lévy jump-diffusion framework that learns conditional drift, diffusion, jump intensity, and size distribution.
result Demonstrates improved calibration, sharper tail control, and risk reduction.
A new method called MCLMC avoids dissipation in sampling from canonical distributions.
problem Sampling from canonical distributions without dissipation.
method Microcanonical Langevin Monte Carlo (MCLMC) as a dissipation-free system of SDE.
result MCLMC converges faster than HMC for lattice φ^4 models.
MESSY estimation recovers symbolic density functions from samples using maximum entropy.
problem Estimating probability density functions from limited samples.
method Maximum-Entropy approach with gradient flow and symbolic regression.
result Efficiently finds optimal symbolic expressions for unknown distributions.
Protocol diagnoses neural HJB-PIDE solvers for Lévy jumps, revealing a missing factor in their importance-proposal density.
problem Neural PDE solvers can match scalar diagnostics but miscompute operators, leading to systematic errors.
method Five-step diagnostic protocol decomposes neural solve into components, compares them with independent reference solutions.
result Corrected a missing 1/2-mixture factor in the neural method's importance-proposal density, improving control accuracy.