arXiv subjects

Tobias Bischoff

Publications and source records attributed to Tobias Bischoff.

12 recordsLinked to original sources

arXiv2026

Statistical Parameter Calibration via the Generalized Fluctuation Dissipation Theorem and Generative Modeling

We introduce a response-theoretic framework that recasts parameter calibration of ergodic stochastic differential equations as a fluctuation-dissipation problem. Our central result is that the full Jacobian of any stationary observable with respect to drift and diffusion parameters admits an exact linear-response representation as a time-correlation integral evaluated along unperturbed dynamics alone, without perturbed simulations, adjoint derivations, or tangent-linear models. The key idea is to interpret infinitesimal parameter variations as causal perturbations of the dynamics, thereby bringing the Generalized Fluctuation-Dissipation Theorem to inverse problems. The resulting kernels couple observables to the score of the invariant density, for which modern score-estimation methods provide practical non-Gaussian estimators. We validate the framework across a hierarchy of models, from analytically tractable processes to stochastic parameterization in the chaotic Lorenz-96 system, and show that a single baseline trajectory can recover parameter sensitivities and calibration updates with accuracy comparable to finite-difference approaches. More broadly, the framework opens a new route to model calibration, statistical inverse problems, and uncertainty quantification when the quantities of interest are long-time statistics of complex dynamical systems.

nlin.CD↗

arXiv2026

Predictor-Driven Diffusion for Spatiotemporal Generation

Multiscale spatial structure complicates temporal prediction because small-scale spatial fluctuations influence large-scale evolution, yet resolving all scales is often intractable. Standard diffusion models do not address this problem effectively since they apply uniform decay across all Fourier modes. We propose Predictor-Driven Diffusion, a framework that combines renormalization-group-based spatial coarse-graining with a path-integral formulation of temporal dynamics. The forward process applies scale-dependent Laplacian damping together with additive noise, producing a hierarchy of coarse-grained fields indexed by diffusion scale $λ$. Training minimizes the Kullback-Leibler divergence between data-induced and predictor-induced path densities, leading to a simple regression loss on temporal derivatives. The resulting predictor captures how eliminated small-scale components statistically influence large-scale evolution. A key insight is that the same predictor provides a path score for reverse-$λ$ sampling, unifying simulation, unconditional generation, and super-resolution in a single framework. Our unified approach is validated through experiments on two multiscale turbulent systems.

physics.flu-dyn↗

arXiv2025

KGMM: A K-means Clustering Approach to Gaussian Mixture Modeling for Score Function Estimation

We propose a hybrid method for accurately estimating the score function, i.e., the gradient of the log steady-state density, using a Gaussian Mixture Model (GMM) in conjunction with a bisecting K-means clustering step. Our approach, which we call KGMM, offers a systematic way to combine statistical density estimation with a neural-network-based interpolation of the score, leveraging the strengths of both. We demonstrate its ability to accurately reconstruct the long-time statistical properties of several paradigmatic systems, including potential systems, chaotic Lorenz-type models, and the Kuramoto-Sivashinsky equation. Numerical experiments show that KGMM yields robust estimates of the score function, even for small values of the covariance amplitude in the GMM, where standard GMM methods tend to fail because of noise amplification. We compare the performance of KGMM against the conventional Denoising Score Matching (DSM) approach, demonstrating that KGMM achieves more faithful reconstruction of the steady-state distribution for low-dimensional systems at a fraction of the computational cost. These accurate estimates allow us to build effective stochastic reduced-order models that reproduce the invariant measures of the target dynamics.

nlin.CD↗

arXiv2025

Surface to Seafloor: A Generative AI Framework for Decoding the Ocean Interior State

Understanding subsurface ocean dynamics is essential for quantifying oceanic heat and mass transport, but direct observations at depth remain sparse due to logistical and technological constraints. In contrast, satellite missions provide rich surface datasets-such as sea surface height, temperature, and salinity-that offer indirect but potentially powerful constraints on the ocean interior. Here, we present a probabilistic framework based on score-based diffusion models to reconstruct three-dimensional subsurface velocity and buoyancy fields, including the energetic ocean eddy field, from surface observations. Using a 15-level primitive equation simulation of an idealized double-gyre system, we evaluate the skill of the model in inferring the mean circulation and the mesoscale variability at depth under varying levels of surface information. We find that the generative model successfully recovers key dynamical structures and provides physically meaningful uncertainty estimates, with predictive skill diminishing systematically as the surface resolution decreases or the inference depth increases. These results demonstrate the potential of generative approaches for ocean state estimation and uncertainty quantification, particularly in regimes where traditional deterministic methods are underconstrained or ill-posed.

physics.geo-ph↗

arXiv2025

Reduced-Order Modeling of Cyclo-Stationary Time Series Using Score-Based Generative Methods

Many natural systems exhibit cyclo-stationary behavior characterized by periodic forcing such as annual and diurnal cycles. We present a data-driven method leveraging recent advances in score-based generative modeling to construct reduced-order models for such cyclo-stationary time series. Our approach accurately reproduces the statistical properties and temporal correlations of the original data, enabling efficient generation of synthetic trajectories. We demonstrate the performance of the method through application to the Planet Simulator (PlaSim) climate model, constructing a reduced-order model for the 20 leading principal components of surface temperature driven by the annual cycle. The resulting surrogate model accurately reproduces the marginal and joint probability distributions, autocorrelation functions, and spatial coherence of the original climate system across multiple validation metrics. The approach offers substantial computational advantages, enabling generation of centuries of synthetic climate data in minutes compared to weeks required for equivalent full model simulations. This work opens new possibilities for efficient modeling of periodically forced systems across diverse scientific domains, providing a principled framework for balancing computational efficiency with physical fidelity in reduced-order modeling applications.

nlin.CD↗

arXiv2025

Gradient-Free Score-Based Sampling Methods with Ensembles

Recent developments in generative modeling have utilized score-based methods coupled with stochastic differential equations to sample from complex probability distributions. However, these and other performant sampling methods generally require gradients of the target probability distribution, which can be unavailable or computationally prohibitive in many scientific and engineering applications. Here, we introduce ensembles within score-based sampling methods to develop gradient-free approximate sampling techniques that leverage the collective dynamics of particle ensembles to compute approximate reverse diffusion drifts. We introduce the underlying methodology, emphasizing its relationship with generative diffusion models and the previously introduced Föllmer sampler. We demonstrate the efficacy of the ensemble strategies through various examples, ranging from low- to medium-dimensionality sampling problems, including multi-modal and highly non-Gaussian probability distributions, and provide comparisons to traditional methods like the No-U-Turn Sampler. Additionally, we showcase these strategies in the context of a high-dimensional Bayesian inversion problem within the geophysical sciences. Our findings highlight the potential of ensemble strategies for modeling complex probability distributions in situations where gradients are unavailable.

cs.LG↗

arXiv2024

Response Theory via Generative Score Modeling

We introduce an approach for analyzing the responses of dynamical systems to external perturbations that combines score-based generative modeling with the Generalized Fluctuation-Dissipation Theorem (GFDT). The methodology enables accurate estimation of system responses, including those with non-Gaussian statistics. We numerically validate our approach using time-series data from three different stochastic partial differential equations of increasing complexity: an Ornstein-Uhlenbeck process with spatially correlated noise, a modified stochastic Allen-Cahn equation, and the 2D Navier-Stokes equations. We demonstrate the improved accuracy of the methodology over conventional methods and discuss its potential as a versatile tool for predicting the statistical behavior of complex dynamical systems.

physics.data-an↗

arXiv2023

Easing Color Shifts in Score-Based Diffusion Models

Generated images of score-based models can suffer from errors in their spatial means, an effect, referred to as a color shift, which grows for larger images. This paper investigates a previously-introduced approach to mitigate color shifts in score-based diffusion models. We quantify the performance of a nonlinear bypass connection in the score network, designed to process the spatial mean of the input and to predict the mean of the score function. We show that this network architecture substantially improves the resulting quality of the generated images, and that this improvement is approximately independent of the size of the generated images. As a result, this modified architecture offers a simple solution for the color shift problem across image sizes. We additionally discuss the origin of color shifts in an idealized setting in order to motivate the approach.

cs.LG↗

arXiv2023

Unpaired Downscaling of Fluid Flows with Diffusion Bridges

We present a method to downscale idealized geophysical fluid simulations using generative models based on diffusion maps. By analyzing the Fourier spectra of images drawn from different data distributions, we show how one can chain together two independent conditional diffusion models for use in domain translation. The resulting transformation is a diffusion bridge between a low resolution and a high resolution dataset and allows for new sample generation of high-resolution images given specific low resolution features. The ability to generate new samples allows for the computation of any statistic of interest, without any additional calibration or training. Our unsupervised setup is also designed to downscale images without access to paired training data; this flexibility allows for the combination of multiple source and target domains without additional training. We demonstrate that the method enhances resolution and corrects context-dependent biases in geophysical fluid simulations, including in extreme events. We anticipate that the same method can be used to downscale the output of climate simulations, including temperature and precipitation fields, without needing to train a new model for each application and providing a significant computational cost savings.

cs.LG↗

arXiv2016

Unsupervised Discovery of El Nino Using Causal Feature Learning on Microlevel Climate Data

We show that the climate phenomena of El Nino and La Nina arise naturally as states of macro-variables when our recent causal feature learning framework (Chalupka 2015, Chalupka 2016) is applied to micro-level measures of zonal wind (ZW) and sea surface temperatures (SST) taken over the equatorial band of the Pacific Ocean. The method identifies these unusual climate states on the basis of the relation between ZW and SST patterns without any input about past occurrences of El Nino or La Nina. The simpler alternatives of (i) clustering the SST fields while disregarding their relationship with ZW patterns, or (ii) clustering the joint ZW-SST patterns, do not discover El Nino. We discuss the degree to which our method supports a causal interpretation and use a low-dimensional toy example to explain its success over other clustering approaches. Finally, we propose a new robust and scalable alternative to our original algorithm (Chalupka 2016), which circumvents the need for high-dimensional density learning.

stat.ML↗

arXiv2015

Order-disorder transitions in a sheared many body system

Motivated by experiments on sheared suspensions that show a transition between ordered and disordered phases, we here study the long-time behavior of a sheared and overdamped 2-d system of particles interacting by repulsive forces. As a function of interaction strength and shear rate we find transitions between phases with vanishing and large single-particle diffusion. In the phases with vanishing single-particle diffusion, the system evolves towards regular lattices, usually on very slow time scales. Different lattices can be approached, depending on interaction strength and forcing amplitude. The disordered state appears in parameter regions where the regular lattices are unstable. Correlation functions between the particles reveal the formation of shear bands. In contrast to single particle densities, the spatially resolved two-particle correlation functions vary with time and allow to determine the phase within a period. As in the case of the suspensions, motion in the state with low diffusivity is essentially reversible, whereas in the state with strong diffusion it is not.

cond-mat.soft↗

arXiv2015

The stability of stratified spatially periodic shear flows at low Péclet number

This work addresses the question of the stability of stratified, spatially periodic shear flows at low Péclet number but high Reynolds number. This little-studied limit is motivated by astrophysical systems, where the Prandtl number is often very small. Furthermore, it can be studied using a reduced set of "low-Péclet-number equations" proposed by Lignieres [Astronomy & Astrophysics, 348, 933-939, 1999]. Through a linear stability analysis, we first determine the conditions for instability to infinitesimal perturbations. We formally extend Squire's theorem to the low-Péclet-number equations, which shows that the first unstable mode is always two-dimensional. We then perform an energy stability analysis of the low-Péclet-number equations and prove that for a given value of the Reynolds number, above a critical strength of the stratification, any smooth periodic shear flow is stable to perturbations of arbitrary amplitude. In that parameter regime, the flow can only be laminar and turbulent mixing does not take place. Finding that the conditions for linear and energy stability are different, we thus identify a region in parameter space where finite-amplitude instabilities could exist. Using direct numerical simulations, we indeed find that the system is subject to such finite-amplitude instabilities. We determine numerically how far into the linearly stable region of parameter space turbulence can be sustained.

physics.flu-dyn↗