Across hydrodynamics, ecology, neuroscience, network dynamics, non-Hermitian physics, and socio-economic systems, asymptotically stable dynamics can exhibit large transient amplifications that are invisible to eigenvalue-based analyses. The mechanism is geometric rather than spectral: perturbations entering along one direction may be expressed transiently along another, allowing asymptotic decay to coexist with strong transient or noise-driven amplification. We introduce non-normal directional response inference, a data-driven method for detecting this geometry from multivariate time series when the governing operator is unknown. A local linear operator is estimated from sliding windows and projected onto the dominant two-dimensional input-response subspace. The reduced dynamics are summarized by the eigenvalue splitting $\Delta$, eigenvector non-orthogonality $K$, and the scale-free ratio $R=K/K_c(\Delta)$, where $K_c(\Delta)$ is the two-dimensional threshold for transient amplification. Controlled benchmarks show that the reduced geometry, particularly $R$, can be recovered from finite data even when the full high-dimensional operator is poorly estimated. Tests across sample size, dimension, training horizon, spectral structure, and non-stationarity confirm that the relevant response geometry requires far fewer observations than full-matrix recovery. Applied in moving windows to electrohysterogram, seizure EEG, freezing-of-gait, and unstable push-up inertial recordings, the method reveals systematic changes around known physiological or behavioral episodes through shifts in $R$, changes in $\Delta$, or stronger projection of fluctuations onto the inferred response direction. It thus exposes interpretable changes in local response geometry without framing the problem as supervised event detection.
We study the reconstruction of an unknown dynamical system from a single noisy scalar time series. The goal is to recover the underlying dynamics for forecasting. We introduce a method that uses differential embedding coordinates to identify a rational closure of the embedding dynamics directly from data. The closure is identified through a weak-form regression pipeline, which avoids unstable pointwise differentiation of noisy data. When applied to noise-free Lorenz and R\"ossler systems, the method recovers closures that support long forecasts across a broad ensemble of realizations ($18.1$ and $7.1$ Lyapunov times respectively). Under $15$--$30\%$ additive Gaussian noise, performance becomes system-dependent. For the Lorenz system, forecast horizons remain short even in the best cases, whereas the R\"ossler system generally performs better in absolute terms, though not once normalized by the Lyapunov time. Our proposed method recovers directly interpretable closure coefficients which we compared against the known analytic closures of the Lorenz and R\"ossler systems.
Causal discovery at large spatiotemporal scale is difficult: variables are physically embedded, candidate interactions grow quadratically with system size, and causal structure changes with system state. We introduce GeoDCD, a geometry-aware neural framework that uses spatial coordinates to initialize a learnable hierarchy and converts a trained nonlinear predictor into time-varying directed graphs through input-output Jacobian sensitivity analysis. On chaotic Lorenz-96 dynamics, GeoDCD attains an F1 score of 0.99, reducing structural Hamming distance by 36.8% relative to the strongest neural baseline, and still leads flat baselines when coordinates are uninformative. Runtime scales approximately linearly over the evaluated range, enabling discovery on a 10,512-node sea-level-pressure grid. Applied to observations, GeoDCD identifies circulation-consistent gateways, resolves El Nino/La Nina-dependent reorganization, and separates energy-to-traffic from traffic-to-energy influence in coupled electric-vehicle and road systems. Edges are neural-Granger sensitivities rather than interventional effects, positioning GeoDCD for mechanistic hypothesis generation where interventions are unavailable.
Haoyang Yan, Kaiqi Zhao, Weiping Wang et al.· 0 citations
Abstract.
Population dynamics in fields such as molecular biology, epidemiology, and ecology exhibit highly stochastic and nonlinear behavior. In gene regulatory systems in particular, oscillations and multistability are especially common. Despite this, none of the currently available stochastic models for population dynamics are both accurate and computationally efficient for long-term predictions. A prominent model in this field, the linear noise approximation (LNA), is computationally efficient for tasks such as simulation, sensitivity analysis, and parameter estimation; however, it is only accurate for linear systems and short-time predictions. Other models may achieve greater accuracy across a broader range of systems, but they sacrifice computational efficiency and analytical tractability. This paper demonstrates that, with specific modifications, the LNA can accurately capture nonlinear dynamics in population processes. We introduce a new framework based on center manifold theory, a classical concept from nonlinear dynamical systems. This approach enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar nonlinear dynamical systems. With these modifications, the LNA can achieve accurate long-term simulations without compromising computational efficiency. We apply our methodology to classes of oscillatory and bistable systems and present multiple examples from molecular population dynamics that demonstrate accurate long-term simulations alongside significant improvements in computational efficiency.
Frederick Truman-Williams, G. Minas· SIAM Journal on Applied Dyna...· 0 citations
This manuscript investigates the Cauchy problem for a class of nonlinear integro-differential equations governing anomalous super-diffusive transport in $\mathbb{R}^N$. The linear dynamics are driven by a dual-scale memory kernel whose Laplace transform is sectorial and exhibits distinct power-law asymptotics at high and low frequencies. This super-diffusive structure precludes the infinite regularizing capacity characteristic of classical parabolic theory; consequently, the associated resolvent operator possesses a heavy algebraic tail in Fourier space, acting as a pseudo-differential operator in the H\"ormander class $S^{-2}_{1,0}$ and restricting spatial smoothing. By establishing rigorous $L^q-L^p$ multiplier estimates, the critical Lebesgue threshold $q_c$ for local well-posedness is determined. To demonstrate the sharpness of this threshold, instantaneous norm inflation -- and consequent ill-posedness -- is proven in the supercritical regime $1<q<q_c$. Furthermore, tracking the structural crossover to the long-time relaxation parameter resolves the global asymptotic dynamics. The nonlocal Fujita-type critical exponent $\rho_F$ is identified, and global-in-time existence along with algebraic decay is established for small initial data in intersection spaces, provided the nonlinearity remains supercritical and overcomes the structural algebraic barrier connecting the dual scales. This general framework applies directly to canonical physical models, including Cole-Cole fractional retardation and multi-scale Prabhakar memory.
We study decentralized stochastic gradient tracking over a time-varying network of $N$ agents under a uniform window-mixing condition. Products of $\tau$ consecutive doubly stochastic mixing matrices contract disagreement by a factor $\lambda<1$, although individual matrices need not contract disagreement strictly and individual communication graphs may be disconnected. We construct a time-varying quadratic norm that turns this window contraction into an exact one-step Lyapunov identity. This leads to coupled one-step recursions for the centroid and disagreement errors, without unrolling the dynamics over communication windows. For smooth strongly convex objectives, the leading stochastic term is $\widetilde{\mathcal O}(1/(NK))$; for smooth convex objectives, it is $\mathcal O(1/\sqrt{NK})$. Both match their centralized mini-batch counterparts and yield linear speedup after a network-dependent transient.