Yongho Sungtaek Ju

Intermittency in turbulence causes some scaling exponents of the longitudinal velocity increment to deviate from Kolmogorov's prediction. Multifractal closures reproduce that deviation accurately, but each fits its parameters to the exponents themselves and leaves the underlying increment distribution unspecified. Using direct numerical simulations of homogeneous isotropic turbulence at Taylor-microscale Reynolds numbers of 433, 611, and 1257, we identify the distribution first. Throughout the inertial range, the increment density is normal-inverse-Gaussian; at the highest Reynolds number, it is preferred over Gaussian, Student-t, and skew-t alternatives at every scale tested. Its parameters are fixed at the dissipative boundary by the kinematic viscosity, the mean dissipation rate, and the universal velocity-gradient skewness and are carried through the inertial range by the measured second and third moments. The exponents then follow without being fitted, the fourth-order exponent falling 0.10 below the measured value and the higher even orders falling short in the same direction. This is expected of a single-shape density whose tails are too thin for the rare increments that dominate high moments. The density asymmetry weakens as the Reynolds number rises. Compared at matched large-scale sampling volume, the change is carried by the increment flatness and not by the skewness, whose contribution is unresolved in every matched pair over a factor of 2.9 in the Reynolds number, while the scale dependence of the flatness gives an intermittency exponent μ=0.314±0.020. The increment distribution is thus set by measurable small-scale physics, and its asymmetry inherits the Reynolds-number dependence of intermittency.

The turbulent energy cascade is widely modeled as a Markov process in scale, valid above a coherence length of about one Taylor microscale. Using direct numerical simulation results and null surrogates that are Markov by construction, the present work shows that this coherence length is state dependent: quiescent regions of the inertial range recover the canonical value, while intermittent regions require 3 to 4 times the scale separation, and the ordering reverses near the dissipation range. The Markov approximation underlying cascade Fokker-Planck and fluctuation-theorem analyses is therefore more restrictive for the intermittent component of the cascade than has generally been assumed.

Constitutive relations—functions linking a local state variable to a material or system property such as thermal conductivity, hydraulic permeability, or reaction rate constant—appear across the physical, engineering, and life sciences. They must often be inferred from sparse, indirect observations of the field they govern rather than from direct assays. Unlike the well-studied spatially varying parameter ϕ(x), the scalar state-dependent function ϕ(u):R→R+ of a single local state variable has received no systematic Bayesian treatment, and a basic question remains open: how much can sparse indirect observations actually inform ϕ(u)? We show the answer is governed by a two-factor attenuation: information must pass first through the inverse of the linearized partial differential equation (PDE) Jacobian and then through the sparse observation mask. The product of these factors lies far below unity for practical configurations, leaving the posterior structurally prior-dominated regardless of the number of indirect observations of the same type and configuration. The primary determinant is not the PDE class but the coupling mechanism—whether ϕ(u) enters the residual through a spatial gradient (as for diffusion coefficients) or pointwise (as for source terms). We instantiate the analysis in a sparse variational Gaussian process with an explicit, checkable prior and a closed-form Laplace posterior, and validate it across four PDE classes, four constitutive shapes, and two prior conditions. A small number of direct constitutive measurements at well-placed state values, bypassing the PDE entirely, collapse the posterior where arbitrarily many indirect observations of the same type cannot.

Parameter identification in nonlinear dynamical systems is complicated by model-form uncertainty arising from systematic biases that violate the zero-mean error assumption of standard data assimilation methods. Recent neural-network-based approaches learn arbitrary bias corrections online. However, they require careful regularization to ensure unique solutions and carry computational overhead from ensemble propagation and in-situ training. We present a framework that integrates the parametrized-background data-weak formulation with attention-based parameter identification networks. It projects model error onto a dictionary of physically motivated spatial templates rather than learning arbitrary corrections. This provides uniqueness through hard subspace constraints rather than soft regularization penalties, albeit at the cost of restricting the representable bias space to patterns anticipated from domain knowledge. The framework is demonstrated on parameter estimation in the Rijke tube model, achieving robust generalization to out-of-distribution (OOD) bias patterns not seen during training. Ablation studies confirm the importance of state supervision for amplitude-sensitive parameters and temporal bias modeling for phase-sensitive parameters. Performance degrades gradually when bias patterns lie outside the template span. This indicates that the attention encoder extracts parameter information from bias-invariant features rather than relying critically on template-based bias capture. The physics-constrained and universal approximator approaches represent complementary points on the flexibility–efficiency tradeoff, suited to different operational contexts.

Solutions to the phonon Boltzmann transport equation under the relaxation-time approximation (RTA) are fundamentally limited in that they do not account for the off-diagonal elements of the scattering matrix, which encode intermode energy redistribution. We find that the phonon in-scattering matrix is globally incompressible, requiring nearly its full rank for any useful Frobenius accuracy. The incompressibility worsens as the Brillouin zone is refined. We show that, despite this difficulty, one can develop a computationally efficient 3D BTE solver incorporating the complete scattering matrix by leveraging our two structural discoveries: the non-equilibrium phonon distribution inhabits a remarkably low-dimensional subspace of mode space regardless of how many phonon modes exist, and the leading singular modes of the scattering operator align selectively with this transport-active subspace. Consequently, truncation incurs negligible transport error even under large norm error. The phonon streaming operator’s mode-diagonal character further motivates a hybrid architecture that exploits these two properties. When applied to nanoscale structures emulating a fin field-effect transistor, our BTE solver quantifies a geometry-independent multiplicative correction to the temperature rise predicted under RTA. Our theoretical work and resulting BTE solver help enable rigorous study of phonon transport and systematic design of devices and structures in the ballistic and quasi-ballistic phonon transport regime.

Kolmogorov-Arnold Networks have emerged as interpretable alternatives to traditional multi-layer perceptrons. However, standard implementations lack principled uncertainty quantification capabilities essential for many scientific applications. We present a framework integrating sparse variational Gaussian process inference with the Kolmogorov-Arnold topology, enabling scalable Bayesian inference with computational complexity that is in theory quasi-linear in sample size. Through analytic moment matching, we propagate uncertainty through deep additive structures while maintaining interpretability. We use three example studies to examine the framework’s ability to distinguish aleatoric from epistemic uncertainty: calibration of heteroscedastic measurement noise in scalar field prediction, quantification of prediction confidence degradation in multi-step forecasting of advection-diffusion dynamics, and out-of-distribution detection in convolutional autoencoders. These results suggest Sparse Variational Gaussian Process Kolmogorov-Arnold Networks (SVGP-KANs) as a promising architecture for uncertainty-aware learning in scientific machine learning.

Radiative transfer in absorbing-scattering media requires solving a transport equation across a spectral domain with 10 5-10 6 molecular absorption lines. Line-by-line (LBL) computation is prohibitively expensive, while existing approximations sacrifice spectral fidelity. We show that the Young-measure homogenization framework produces solution tensors I(xi, µm, sj) that admit low-rank tensor-train (TT) decompositions whose bond dimensions remain bounded as the spectral resolution Ns increases. Using molecular line parameters from the HITRAN database for H2O and CO2, we demonstrate that: (i) the TT rank saturates at r =8 (at tolerance e = 10-6) from Ns = 16 to 4096, independent of single-scattering albedo, Henyey-Greenstein asymmetry, temperature, and pressure; (ii) quantized tensortrain (QTT) representations achieve sub-linear storage scaling; (iii) in a controlled comparison using identical opacity data and transport solver, the homogenized approach achieves over an order of magnitude lower L2 error than the correlated-k distribution at equal cost; and (iv) for atomic plasma opacity (aluminum at 60 eV, TOPS database), the TT rank saturates at r = 15 with fundamentally different spectral structure (bound-bound and bound-free transitions spanning 12 decades of dynamic range), confirming that rank boundedness is a property of the transport equation rather than any particular opacity source. These results establish that the spectral complexity of radiative transfer has a finite effective rank exploitable by tensor decomposition, complementing the spatial-angular compression achieved by existing TT and dynamical low-rank approaches.

Tensor-train methods compress transport equations by orders of magnitude in benchmark calculations, but they largely lack a predictive theory: reported rank structures are empirical, measured a posteriori on specific problem instances, and no failure modes of quantized tensor-train (QTT) decomposition have been systematically identified for transport problems. This work develops a predictive framework for tensor-train rank behavior using two-dimensional radiative transfer with spectrally resolved opacity as a vehicle. We report three principal findings. First, we identify a fundamental operator--solution rank asymmetry under uniform QTT decomposition: spatial operator ranks remain bounded, but spatial solution ranks inflate as O(Nx) because the multivariate coupling of the transport solution lacks the algebraic separability that makes QTT effective for operators. Selective quantization, applying QTT only to the spectral dimension, resolves this asymmetry and achieves O(log Nσ) storage without spatial penalty. Second, combining this approach with Young-measure homogenization, we establish the first a priori spectral rank guarantee for tensor methods in radiative transfer equations: the spectral band rank is universally bounded at 2--3, traced to the two-mode structure of the exponential attenuation factor. Because the spectral and spatial--angular compression mechanisms act on disjoint tensor modes, their ratios compose multiplicatively, exceeding 104 at production scales. Third, the framework is validated on self-consistent collisional-radiative transport with atomic plasma opacity, where it closes the multiphysics coupling loop with six-digit accuracy---further demonstrating that tensor-compressed solvers can support real multiphysics workflows. The present work helps establish tensor-train methods for transport equations as not just a compression technique but a framework whose effectiveness can be predicted from the structure of the underlying physics.