New method models stochastic systems efficiently from data.
problem Modeling complex spatial interactions from data.
method Cox process representation and machine learning.
result Efficient algorithm for parameter inference and model selection.
Solves probabilistic Lambert problem connecting astrodynamics with optimal mass transport.
problem Determining spacecraft velocity for given positions with probabilistic constraints.
method Generalized optimal mass transport (OMT) and Schrödinger bridge (SBP) connections.
result Existence and uniqueness of solution for probabilistic Lambert problem.
Schrödinger bridge solved with Weyl calculus for quadratic state cost.
problem Optimal control policy to steer joint state statistics.
method Weyl calculus in quantum mechanics for reaction-diffusion PDEs.
result Explicit Markov kernel for quadratic state cost found.
Study shows neural operators can efficiently solve complex reaction-diffusion systems.
problem Efficiently solving nonlinear reaction-diffusion systems using neural operators.
method Laplacian-based neural operators applied to a generalized Gierer-Meinhardt system.
result Explicit approximation error bounds established for neural operators in terms of network parameters.
New method uses vortex strings to untangle knots.
problem Untangling complex knots.
method Reaction-diffusion dynamics of vortex strings in a nonlinear PDE.
result Evolution preserves knot topology and untangles unknots.
Study proves existence of weak solutions for volume-preserving mean curvature flow using phase field method.
problem Existence of weak solutions for volume-preserving mean curvature flow.
method Phase field method applied to reaction-diffusion equation with nonlocal term.
result Existence of weak solutions proved for the volume-preserving mean curvature flow.
Researchers prove inequalities for reaction-diffusion systems using a new curvature-dimension condition.
problem Proving Li-Yau and Harnack inequalities for systems of linear reaction-diffusion equations.
method Introducing a hybrid curvature-dimension condition and proving a differential Harnack estimate.
result A Harnack inequality holds under the hybrid curvature-dimension condition CDhyb(0,d) with d<∞. PASTIS selects minimal models from stochastic dynamics data.
problem Overfitting in model selection for stochastic dynamics.
method Combining likelihood-estimation statistics with extreme value theory.
result PASTIS reliably identifies minimal models, even with low sampling rates or error.
GF-Net learns Green's functions for linear reaction-diffusion equations.
problem Learning Green's functions for linear reaction-diffusion equations on arbitrary domains.
method GF-Net, a neural network, learns Green's functions in an unsupervised manner using physics-informed approach and symmetry.
result GF-Net efficiently solves linear reaction-diffusion equations under various boundary conditions and sources.
This study explains how adversarial interaction creates non-homogeneous patterns using a pseudo-Reaction-Diffusion model.
problem Understanding how adversarial interaction leads to non-homogeneous patterns in systems.
method Developed a pseudo-Reaction-Diffusion model to explain the mechanism.
result Turing instability is involved in creating non-homogeneous patterns.
Global solutions and smoothing effects for reaction-diffusion equations on manifolds.
problem Global existence and smoothing effects for reaction-diffusion equations on Riemannian manifolds.
method Functional analytic methods, Sobolev and Poincaré inequalities.
result Existence of global solutions under certain conditions on the manifold.
Global solutions found for certain reaction-diffusion equations on specific manifolds.
problem Understanding reaction-diffusion equations on various manifolds.
method Analyzing the bottom of the L2 spectrum of −Δ and using time-independent nonlinearities. result Global existence of solutions for certain power nonlinearities on specific manifolds.
Physically-inspired Gaussian process models study post-transcriptional regulation in Drosophila.
problem Understanding spatiotemporal interactions between mRNAs and gap proteins in post-transcriptional regulation.
method Two physically-inspired Gaussian process models based on reaction-diffusion equations, tested with mRNA expression data.
result Novel GP model requires only kernel function differentiation, simplifying spatial discretisation.
We generalize the reaction-diffusion model A + B -> 0 in order to study the impact of an excess of A (or B) at the reaction front. We provide an exact solution of the model, which shows that linear response breaks down: the average displacement of the reaction front grows as the square-root of the imbalance. We argue t…
Global existence and smoothing effects for reaction-diffusion equations with blowup in infinite time.
problem Analyzing reaction-diffusion equations with power-type nonlinearity and slow diffusion.
method Functional analytic methods based on Sobolev and Poincaré inequalities.
result Solutions corresponding to large initial data blow up everywhere in infinite time on Cartan-Hadamard manifolds.
In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a family of smooth curv…
Several models of stock trading [P. Bak et al, Physica A {\bf 246}, 430 (1997)] are analyzed in analogy with one-dimensional, two-species reaction-diffusion-branching processes. Using heuristic and scaling arguments, we show that the short-time market price variation is subdiffusive with a Hurst exponent H=1/4. Biase…
Fractional reaction-diffusion model explains financial market dynamics.
problem Reproduce realistic price dynamics in financial markets.
method Proposes a fractional reaction-diffusion model with heterogeneous agent frequencies.
result Impact kernel decays as t−1/2 in the diffusive case, inconsistent with market efficiency; β can be tuned to match empirical values. New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
We solve a Schrödinger bridge with a quadratic state cost, finding a closed-form solution.
problem Optimizing diffusion processes between given distributions.
method Regularized Schrödinger bridge with a quadratic state cost.
result Closed-form solution for the Markov kernel of the regularized Schrödinger bridge.
INP accelerates stochastic simulations using deep Bayesian active learning.
problem Computational expense of stochastic simulations at fine-grained resolution.
method Interactive Neural Process (INP) framework combining spatiotemporal surrogate model and active learning acquisition function.
result STNP outperforms baselines in accelerating stochastic simulations and LIG achieves state-of-the-art for Bayesian active learning.
In this note, we propose Bernstein's problem and De Giorgi's conjecture for spatially inhomogeneous equations, as well as De Giorgi's conjecture for system of reaction-diffusion equations.
The paper proves stability of a blowup solution for Yang-Mills heat flow.
problem Stability of blowup solutions for Yang-Mills heat flow.
method Small perturbation analysis and explicit self-similar blowup solution.
result Stability of the explicit self-similar blowup solution under perturbations.
Study nonlocal models for curve dynamics, linking to crossing number complexity.
problem Understanding the motion and complexity of embedded curves using nonlocal models.
method Analyzes a family of nonlocal, energy-driven dynamical models for closed, embedded curves.
result Establishes a connection between the studied models and a generalized crossing number.
Unified analytic account of correlation emergence and Epps effect in coupled limit order books
problem Correlation emergence and Epps effect in coupled limit order books
method Discrete random-walk description of order flow with creation, cancellation, and diffusion, coupled reaction-diffusion equations with moving reaction boundary
result Realized correlations as a function of aggregation time
RAD estimates gradients with less memory, faster than small batch sizes.
problem Training deep models with stochastic gradient descent requires exact gradients, but they are not needed.
method Developed a framework for randomized automatic differentiation (RAD) to compute unbiased gradient estimates with reduced memory.
result RAD converges in fewer iterations than using a small batch size for feedforward networks and similar number for recurrent networks.
Method learns latent dynamics of complex systems from noisy data.
problem Challenging to construct ROMs from noisy high-dimensional data.
method Recurrent stochastic variational deep kernel learning (SVDKL).
result Framework accurately predicts system evolution in low-dimensional latent spaces.
Extended model accounts for finite memory effects in financial markets.
problem Modeling latent liquidity and its impact in financial markets.
method Continuous reaction-diffusion setup with finite cancellation and deposition rates.
result Square root impact law with finite memory corrections and linear permanent impact.
We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuatio…
New PINN architectures learn high-frequency features using Fourier features.
problem PINNs struggle with high-frequency or multi-scale features.
method Employ spatio-temporal and multi-scale random Fourier features.
result Effective PINN models for multi-scale PDEs.
New methods solve complex equations using neural networks.
problem Long-time integration of nonlinear stochastic PDEs.
method Physics-Informed Neural Networks (PINNs) with dynamically orthogonal (DO) and bi-orthogonal (BO) constraints.
result Overcomes limitations of original DO/BO methods and can handle inverse problems.
Method predicts multistable system states from sparse measurements.
problem Predicting multistable system states from limited data.
method Semi-supervised classification with SPML optimization.
result 95% accuracy in predicting reaction-diffusion equation states.
Framework uses optimal transport for neural architecture search.
problem Optimizing neural architectures in deep learning.
method Semi-discrete optimization using optimal transport.
result Gradient flow and minimizing movement scheme converge to reaction-diffusion equations.
StatFEM uses low-rank approximations to scale Bayesian statFEM for high-dimensional problems.
problem Model misspecification and scalability in high-dimensional physical systems.
method Low-rank approximation of covariance matrix, Bayesian filtering, sparse data reconstruction.
result Reconstructs sparsely observed data-generating processes with minimal loss of information.
New numerical methods for pricing American options using semilinear BSDEs.
problem Pricing American options with complex payoff functions in multi-dimensional settings.
method Proposed two numerical schemes based on branching processes and randomization.
result Simple randomization provides good results for approximating discontinuous drivers.
The paper proves a Pohozaev identity for warped product solitons with bounded curvature.
problem Characterizing warped product metrics in terms of their curvature and base dimensions.
method Proving a Pohozaev-type identity for warped product solitons with bounded curvature.
result Warped product metrics with bounded curvature and 2-dimensional base must be cross products.
The paper solves the Cauchy problem for Friedrichs systems on specific spacetime manifolds.
problem Investigating the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundaries.
method Admissible boundary conditions are imposed to show the existence and uniqueness of strong solutions. For hyperbolic systems, the Cauchy problem is also well-posed in the Hadamard sense.
result Existence and uniqueness of strong solutions for the Cauchy problem are proven under admissible boundary conditions.
A microscopic approach to macroeconomic features is intended. A model for macroeconomic behavior under heterogeneous spatial economic conditions is reviewed. A birth-death lattice gas model taking into account the influence of an economic environment on the fitness and concentration evolution of economic entities is nu…
DeepONets combine neural networks with physics constraints for PDEs and parameter estimation.
problem Estimating parameters in PDEs with uncertainty quantification.
method Physics-informed neural networks (PINNs) integrated with Deep Operator Networks (DeepONets) for Bayesian inference.
result Robust and accurate solutions with comprehensive uncertainty quantification.
Develops a method for identifying structured dynamical systems from data.
problem Identifying structured dynamical systems from undersampled and noisy data.
method Sparse least-squares fitting via ℓ1−ℓ2 optimization with the alternating direction method of multipliers. result The method is stable and successful under certain conditions, as shown by theoretical guarantees and computational results.
GD-VAEs learn dynamics from observations using geometric and topological information.
problem Learning parsimonious representations of nonlinear dynamics from observations.
method Develops data-driven methods incorporating geometric and topological information using Variational Autoencoders (VAEs).
result GD-VAEs provide methods for learning reduced dimensional representations of nonlinear dynamics.
ANODEV2 extends Neural ODEs to include evolving parameters.
problem Training and accuracy of neural networks.
method Coupled ODE-based framework for evolving neural network parameters.
result ANODEV2 achieves higher accuracy than baseline models and Neural ODEs.
New method solves high-dimensional Bayesian inverse problems efficiently.
problem Efficiently solving high-dimensional Bayesian inverse problems with limited data.
method Physics-informed Neural Operators with RealNVP architecture for invertibility and differentiability.
result Accurate approximations of the full posterior without additional forward solves or sampling.
New method extracts hidden phases in binary mixtures using tubular tilings.
problem Hidden phases in binary mixtures are difficult to observe.
method Introduce tubular tilings for discretizing binary mixtures on smooth manifolds.
result Recover topological information about hidden phases from observable phases and interfaces.
Method improves simulation accuracy by mitigating distribution shift in hybrid systems.
problem Mitigating distribution shift in machine-learning augmented hybrid simulation.
method Tangent-space regularized estimator to control distribution shift.
result Marked improvements in simulation accuracy, especially for systems with high distribution shift.
A machine learning method to discover physical theories from data.
problem Discovering explicit analytic equations from physical data.
method Iterative machine learning approach using random combinations of highly correlated expressions.
result Extracting explicit analytic equations from physical data.
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.
Compositional diffusion models simulate coupled PDEs efficiently.
problem Efficiently simulating long-horizon coupled PDE systems.
method Diffusion models trained on decoupled data are composed at inference time.
result Compositional diffusion models recover coupled trajectories with low error.