By virtue of the sheer compute demands of large-scale artificial intelligence (AI) and high-performance computing (HPC), modern multi-core microprocessors suffer from severe power density limits and thermal throttling, entailing an urgent need for principled methods for predictive, physics-informed dynamical thermal management (DTM). Here, we introduce a hardware/software co-design physics-informed PDE-constrained DTM throttling optimal control algorithm, ADACT (Adjoint-Driven Adaptive Co-Design Thermal Control), that minimizes thermal risk and cooling energy overhead while maximizing core execution throughput. The underlying physical temperature field is governed by a two-dimensional nonhomogeneous parabolic reaction-diffusion equation with spatially localized, time-varying power source terms and variable convective cooling subject to nonhomogeneous Dirichlet boundary conditions. We employ a discretization of the continuous state PDE via a Crank-Nicolson finite difference scheme with a 5-point discrete Laplacian operator on a uniform spatial mesh, which preserves unconditional numerical stability and $O(Δx^2 + Δy^2 + Δt^2)$ second-order temporal and spatial accuracy. To efficiently compute exact gradients, we construct and implement a continuous backward-in-time adjoint state method with L-BFGS as our optimization algorithm of choice. The accuracy and second-order convergence rates of both forward state and backward adjoint solvers are quantitatively verified via the Method of Manufactured Solutions (MMS), and gradient accuracy is validated using directional derivative Taylor tests. Numerical experiments demonstrate that ADACT dynamically throttles software power workloads and modulates hardware cooling efforts to strictly uphold physical thermal thresholds while maintaining operational throughput.
Predictions with dynamical systems can be computationally expensive---even with reduced models---because reaching future prediction times can require many sequential time integration steps. We propose Flow-Map First-Order Trajectory Matching (FM-FTM), a nonintrusive model reduction approach that accelerates ensemble predictions of stochastic dynamical systems by learning flow maps that replace multiple time-integration steps. Our method uses first-order trajectory matching (FTM) to learn a deterministic representation of the stochastic probability-current dynamics and then reduces its evolution over time into a flow map whose evaluation directly yields predictions over long time horizons. Numerical experiments on evolution problems with stochastic forcing demonstrate that FM-FTM accelerates ensemble predictions by up to two orders of magnitude while accurately predicting time marginals and ensemble statistics.
Alberto De Marchi, Yura Malitsky, Adrien B. Taylor
We derive convergence guarantees for mirror descent and proximal mirror descent with a logarithmic barrier as the distance-generating function. When the solution lies on the boundary, the Bregman divergence blows up and classical guarantees become vacuous. While existing remedies recover $O(\log k/k)$ rates by adapting classical bounds, we propose a more direct route in which the Bregman distance to the solution never appears, and which achieves the same rate with constant stepsize. We also construct an example showing that this rate cannot be improved in general. Finally, we compare the proposed approach with interior-point methods.
We develop and analyze multilevel gradient-regularized Newton methods for unconstrained convex optimization with Lipschitz-continuous Hessians. By computing Newton corrections in lower-dimensional subspaces, the proposed methods aim to reduce computational costs while preserving the global $\mathcal{O}\left(k^{-2}\right)$ convergence rate in function values of the full-space method. To establish this rate for multilevel methods, we introduce a class of row-nested restriction operators that ensures monotonicity of the reduced-gradient norms across levels. Numerical experiments with row-nested operators demonstrate convergence behavior comparable to full-space gradient-regularized Newton, while offering substantial computational savings on large-scale instances.
Automatic-differentiation-based inverse analysis methods, including physics-informed neural networks (PINNs) and differentiable programming, have recently shown great promise due to their ability to compute accurate gradients and convergence efficiency. However, their applicability to falling weight deflectometer (FWD) backcalculation remains unexplored. This study critically evaluates PINN-based inverse analysis for a multilayer pavement system and investigates differentiable finite element method (DiffFEM) as an alternative based on a synthetic benchmark. The standard PINN does not recover layer moduli because of the sharp domain discontinuities inherent to layered pavement systems. Although we use an extended PINN with domain decomposition (XPINN), which shows better performance on discontinuous domains, its performance remains highly sensitive to loss weighting and network architecture, and degrades under measurement noise. By contrast, DiffFEM consistently achieves more accurate, stable, and computationally efficient inversion results. These results show and quantify the expected advantage of DiffFEM, which enforces the governing physics as a hard constraint in the solver, over PINN-based approaches, where the governing physics is imposed as a soft constraint through the loss function. More broadly, the findings suggest that the choice between PINN- and DiffFEM-based inverse analysis needs careful consideration, with DiffFEM offering practical advantages when an efficient and robust differentiable forward solver is available.
We study sampling and recovery for tensor-valued dynamical systems evolving under the t-product, using partial spatial observations over time. Although the t-product Fourier transform yields slicewise matrix dynamics, recovery does not generally decouple across slices. Eigenvalues shared across Fourier-domain frontal slices couple the corresponding eigenspaces into aggregated spectral blocks, which determine the tensor-specific recovery geometry and space--time tradeoff. We establish deterministic identifiability conditions for the initial tensor state. In the diagonalizable case, these are frame conditions on the aggregated blocks. We extend them to non-diagonalizable systems through generalized eigenspaces and Jordan structure. A temporal--spectral factorization of the finite-horizon stacked recovery system underlying Tensor Dynamical Sampling (DS) yields a decoupled two-stage formulation once the temporal threshold is reached. Under independent spatial subsampling, we prove high-probability recovery guarantees using empirical Gram matrices on the aggregated blocks. The bounds depend on aggregated multiplicities, incoherence, and population conditioning. Stability estimates separate the spatial frame factor from the temporal Vandermonde factor and show that increasing temporal depth may worsen numerical conditioning even after identifiability is ensured. We implement Tensor DS and evaluate it on synthetic and real data. Phase transitions track the predicted blockwise sampling thresholds, and stability experiments illustrate Vandermonde conditioning. Experiments on sea surface temperature and turbulence velocity data show that a learned t-product system operator generally yields more accurate and structurally faithful reconstructions than vectorized low-rank, Tucker-type, graph-regularized, graph-spectral, and vectorized dynamical sampling baselines.
Levenberg-Marquardt power flow (LMPF) addresses ill-conditioned equations and provides least-squares stationary points for infeasible operating requests. Conventional implementations form damped normal equations, where J^T J introduces fill, squares the spectral condition number of the undamped part, and adds sparse-product cost. We propose an augmented LMPF formulation with a symbolic-permutation-based Jacobian operator that fills the augmented blocks directly. We prove that the matrix is symmetric quasi-definite for positive damping, enabling fixed-order, pivot-free QDLDL factorization. This property also motivates the solver choice: KLU value-only refactorization can reuse an unsuitable pivot order, forcing fresh full factorizations on difficult LMPF trajectories. We further introduce one-probe adaptive trust-region initialization for the radius and damping. Direct assembly is 7.5-11.9x faster than the fastest coordinate-list path and 18.0-58.8x faster than cached normal-equation sparse-product formation; QDLDL reduces numerical refactorization and triangular-solve time by up to 3.40x on identical matrices. Adaptive initialization solves flat-started RTE cases in 19-25 factorizations, 7.5-11.6x faster than reproduced three-step LM. On case2383wp, our 84-iteration run performs 84 factorizations and 168 resolves in 237 ms, versus published times of 200.780-441.72 s with fewer outer iterations, corresponding to speedups of 847-1860x. These results require complete factorization and resolve accounting and reveal initialization and legacy-parameter effects. DC-initialized NR exhibits fast quadratic convergence throughout every tested solvable loading range, motivating clearer, reproducible definitions of ill-conditioned power-flow benchmarks.
Score-based generative models (SGMs) need to approximate the scores $\nabla \log p_t$ of the intermediate distributions as well as the final distribution $p_T$ of the forward process. The theoretical underpinnings of the effects of these approximations are still lacking. We find precise conditions under which SGMs are able to produce samples from an underlying (low-dimensional) data manifold $\mathcal{M}$. This assures us that SGMs are able to generate the "right kind of samples". For example, taking $\mathcal{M}$ to be the subset of images of faces, we find conditions under which the SGM robustly produces an image of a face, even though the relative frequencies of these images might not accurately represent the true data generating distribution. Moreover, this analysis is a first step towards understanding the generalization properties of SGMs: Taking $\mathcal{M}$ to be the set of all training samples, our results provide a precise description of when the SGM memorizes its training data.