We include elliptical cross-sections to physically represent people, and irregular polygons to represent wheelchair users, in an anisotropic social force model whose velocity and angular dependence also captures the social tendency for people to avoid walking into one another. Physical interactions are included that depend on the area of overlap between people, or obstacles, to capture normal forces that resist compression and tangential forces that resist sliding motion. The model is further extended to include decision making capabilities, small social groups, the spread of panic, and herding behavior. A large box store is simulated during an evacuation where people move through the store, along the shortest path, to their desired exits. The effects of exit choice, or the perceived availability of exits, on exit times is elucidated. It is found that ignoring 'staff only' exits, and only exiting from the main entrances, can significantly increase average egress times.

This study investigates the predictive capability of a threshold inception-based multiphysics modeling approach for ionic wind fans by direct comparison with experimental measurements. A wire-to-cylinder electroaerodynamic (EAD) fan with variable electrode spacing is used as a reference system to assess the model's ability to reproduce airflow characteristics, discharge current, and performance trends under atmospheric conditions. Numerical simulations show good qualitative agreement with experimental results across all tested configurations; however, systematic deviations emerge at higher voltages and larger electrode gaps. Analysis of these discrepancies indicates that the commonly adopted assumption of perfectly smooth emitter surfaces can limit model accuracy. Experimental characterization of the emitter wire reveals micro-scale surface protrusions, which locally enhance the electric field and alter corona inception behavior. Incorporating representative surface roughness into the numerical model improves quantitative agreement with measured airflow velocities. The results demonstrate that while the threshold inception model provides a robust foundation for EAD fan simulations, electrode surface morphology is a critical factor for reliable prediction. This work advances the validation and refinement of ionic wind fan modeling methodologies and identifies key considerations for the development of more accurate engineering-oriented simulation tools.

Diamond hosts optically active point defects central to quantum technologies, yet the carbon self-interstitials introduced during growth and irradiation compete with them and form new defects whose configurational landscape is poorly charted, as subtle energy differences govern the competing minima and pathways. Here we build an interstitial-focused dataset by active learning and benchmark three machine-learning interatomic potentials -- GAP, NEP and the equivariant MACE -- against density functional theory for energies, forces and migration barriers. MACE reproduces the reference energetics and relative stabilities, whereas the others can misorder the ground states. Annealing molecular dynamics with the validated potentials uncovers a series of previously unreported carbon interstitial clusters, from di- to octa-interstitials -- several introducing in-gap states of interest as colour centres -- and shows that their metastability is governed by kinetically accessible pathways rather than energetic ordering. These results chart the interstitial defect landscape and accelerate defect discovery for quantum technologies.

Neuroevolution Potential (NEP) is one of the most efficient machine-learned interatomic potential frameworks for large-scale atomistic simulations. However, its original training strategy remains computationally demanding, limiting systematic exploration of model architectures and training protocols. Here, we present TorchNEP, a PyTorch-based implementation of NEP that combines analytically derived gradients, adaptive optimization, and a two-stage training strategy. TorchNEP accelerates training by more than two orders of magnitude while maintaining full compatibility with existing NEP models. We further show that the improvement in predictive accuracy primarily originates from the two-stage training protocol rather than the optimization algorithm itself. Across diverse benchmark datasets, TorchNEP consistently improves force and stress predictions while maintaining comparable or improved energy accuracy. Benchmark evaluations on elemental and alloy systems demonstrate enhanced predictive performance for both atomic configurations and key materials properties. Furthermore, we show that increasing model complexity does not necessarily improve predictive performance despite reducing training errors. Overall, TorchNEP provides an efficient and flexible training framework for developing more accurate and robust machine-learned interatomic potentials.

This work follows the evolution of financial networks in Norway over a period of nine years at a monthly rate. The data consist of board directors and their affiliations to companies, which we model as simplicial complexes. In this framework, directors are represented as nodes and companies as faces of the complex. To characterize the latter, we focus on three topological measures: the Euler characteristic, computed through the Betti numbers, torsion computed through the reduced determinant of the higher-order Laplacians, and higher-order clustering coefficients. The first two fail to capture the effect of imposed law on representation, unlike our notion of curvature which is a geometrical measure computed from the coefficients of the series expansion of the heat kernel in powers of time, which is our major contribution in this work. In particular, the Euler characteristic integrates curvature, and thus local information is lost. Subsequently, not every topological measure can reliably capture shocks in networks. Further, the number of spanning trees may undergo significant changes at the lowest order, yet these changes need not be reflected in the torsion. Conversely, the change in the curvature revealed variation in the board interlock due to legislation, and serves as a sensitive measure for detecting shocks in networks. Inflection points in curvature are associated with external forcing, and minima with shock arrival times. Sharp transitions are also observed in the components of torsion, while smooth changes are observed in higher-order clustering.

When stochastic field dynamics are cast into a path-integral formulation, perturbation theory becomes systematic but the resulting expansion quickly grows combinatorially large. The setting targeted here includes multi-component, multi-dimensional fields with matrix propagators, tensor-valued couplings, and non-Gaussian driving noise specified by arbitrary $n$-point cumulants. Wick pairings grow factorially, and component indices must be routed through the tensor-valued vertices. The useful output is not a raw contraction list, but a diagram table: one entry per topology, with multiplicities, coupling sums, signs, and causal constraints resolved. We present sft-wick, an open-source Python package that constructs these diagram tables and computes their integrals numerically. Given an action and an observable, it enumerates topologically distinct Feynman diagrams, derives their algebraic coefficients, and evaluates the resulting diagram integrals from user-supplied response and cumulant functions. The core algorithm enumerates spatial topologies before routing component indices, avoiding contraction-by-contraction Wick expansion. Response-field constraints, including vanishing response-response contractions, the ito prescription, and the absence of causal response loops, are enforced during enumeration. Predictions are validated against direct Langevin simulation, agreeing to within the simulation's statistical noise.

The pulsar magnetosphere has only recently been addressed using Physics-Informed Neural Networks (PINNs), by deploying a domain-decomposition approach and treating the separatrix and equatorial current sheet as infinitesimally thin discontinuities. However, this baseline requires extensive manual hyperparameter tuning, achieves limited final accuracy and demands several hours of training. We refine this framework by introducing domain-specific neural architectures based on Kolmogorov-Arnold networks, an automated adaptive training pipeline and a physics-based convergence criterion that eliminate the need for manual calibration. The proposed methodology delivers self-consistent axisymmetric magnetosphere solutions with mean squared errors of the PDE residuals at O(1e-6) in double precision - an improvement of two orders of magnitude over the baseline - while achieving convergence in under 20 minutes in single precision. Importantly, the method reliably resolves stellar radii reduced by up to 80% compared to the baseline, overcoming the severe spatial scale disparities that also challenge traditional solvers. Furthermore, by varying the flux that opens to infinity, we provide a correction to the equation that connects it to the equatorial T-point's position. The complete framework is released as the open-source library PulsarX.

Mathematicians understand a PDE solution through mathematical structures rather than tables of computed values. Historically, this has been the product of mathematical analysis, carried out by hand for each problem individually. Neither numerical simulation nor neural networks produce those structures directly. We propose Agentic Symbolic Search (ASYS), a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. This makes the search an automated form of inductive-bias injection rather than blind symbolic regression. For problems with known analytical forms, ASYS recovers these forms naturally; for other problems, ASYS constructs analytical approximations which can guide mathematicians toward further analysis. In our experiments, across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, in settings where no closed-form description was previously available. ASYS shows the possibility of a new paradigm for characterizing PDE solutions, beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.

We develop QCPIKAN, the first quantum-classical physics-informed Kolmogorov-Arnold network designed to solve partial differential equations (PDEs). Built upon Chebyshev-polynomial KAN layers and parameterized quantum circuits, this hybrid framework embeds physical constraints into the training loss to enforce physical consistency. Our theoretical investigations grounded in approximation theory prove that this design accelerates high-frequency error convergence to an exponential rate and effectively mitigates numerical dispersion. We validate the framework across three typical seepage scenarios in porous media, including single-phase flow, component transport and two-phase flow. Compared with existing quantum-classical physics-informed neural networks, QCPIKAN achieves superior performance in global prediction accuracy, local error control, dynamic evolution tracking and displacement front localization. This work provides a robust and efficient alternative for solving complex PDEs.

High-quality single-photon sources, realized through saturable emitters, photon blockade, or heralded pair generation, are indispensable building blocks for photonic quantum platforms. Although these mechanisms suppress multiphoton emission through distinct principles typically captured by analytical models, their practical implementation is constrained by conflicting requirements for purity, brightness, and indistinguishability, which must be balanced within high-dimensional design landscapes. Here, we propose a computational framework for optimizing competing metrics of single-photon sources. Building on a Liouville-space adjoint formulation that efficiently evaluates multiple objectives in Markovian open quantum systems, we develop a Jacobian-based update, which ensures first-order monotonic reduction of multi-objective costs. By incorporating simulated annealing to escape gradient-vanishing plateaus, our framework achieves a design success rate of nearly 60 % for photon blockade with g2(0) smaller than 0.1 and theoretically bounded brightness across a broad parameter space, without any analytical guidance. This framework provides a general recipe for multi-objective design of open quantum systems.

We develop a structure-oriented randomized neural network framework, termed SO-RaNN, for the Poisson-Nernst-Planck (PNP) system and the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The decoupled linearized subproblems are solved iteratively by randomized neural networks in a space-time framework. For the concentration variables, a pointwise cut-off is used to enforce positivity at the value level, and discrete mass-scaling factors are computed at selected correction instants and interpolated in time, so as to ensure exact mass matching at those instants and to promote approximate mass preservation between them. To introduce an auxiliary discrete dissipation mechanism, we further employ an SAV-type post-processing correction, which yields monotonicity of the SAV auxiliary variable under the ideal SAV update. For the PNP-NS system, a structure-preserving randomized neural network (SP-RaNN) is used for the velocity field, so that the velocity approximation satisfies the incompressibility constraint pointwise by construction. On the theoretical side, we derive residual-based estimates for the raw, uncorrected RaNN solvers of the linearized subproblems, formulate a conditional local-in-time convergence result for the raw outer Picard iteration of the PNP system, and analyze the value-level positivity correction together with the mass-correction and SAV post-processing steps. For the PNP-NS system, we establish an approximation result for the SP-RaNN space and provide a conditional error statement for the corresponding linearized Oseen-type problem. Numerical experiments demonstrate approximation accuracy in the source-driven manufactured tests and illustrate the intended value-level positivity correction, selected-time mass matching, computed free-energy curves based on the final gauge-fixed potential, and divergence-free approximation in benchmark tests.

We introduce SEA-PINN, a novel architecture that incorporates a Squeeze-Excitation-like attention mechanism into physics-informed neural networks to dynamically recalibrate the importance of neurons across layers. A key feature of SEA-PINN is its highly stable initialization. On 17 out of 20 benchmark problems, SEA-PINN exhibit nearly negligible variance and significantly reduced initial loss, establishing a quasi-deterministic and favorable starting point for optimization. Notably, without employing Fourier feature embeddings or periodic activation functions, SEA-PINN attained competitive accuracy (83\% vs. 90\% improvement relative to FNN-PINN on the high-frequency case 7) as compared with TSA-PINN-a model specifically engineered for high-frequency problems via learnable frequencies in sinusoidal activations. Furthermore, integrating SEA-PINN into TSA-PINN boosted performance by 42.49\%. These results underscore SEA-PINN as a lightweight plug-in module that enhances nonlinear representation power, promotes more robust and efficient convergence, and strengthens the overall reliability of physics-informed learning.

Magnesium ions play an essential role in many biological processes but remain challenging to model in biomolecular simulations. Despite considerable scientific effort, classical force fields fail to simultaneously reproduce key structural, thermodynamic and kinetic solution properties, likely due to their inability to explicitly account for quantum many-body effects. Here, we develop and systematically benchmark MACE neural network potentials (NNPs) for aqueous MgCl$_2$ solutions trained on revPBE-D3/zd and revPBE0-D3/zd density functional theory reference data and assess their ability to reproduce a broad range of experimental solution properties including the structure of the first hydration shell, diffusion coefficient, activity derivative, water-exchange rate and mechanism as well as solvation free energy. Both NNPs accurately reproduce the octahedral structure of the first hydration shell, ion pairing properties and diffusion coefficients. Combining the NNPs with transition path sampling and other enhanced sampling techniques allows us to capture the rare event of water exchange in the first hydration shell of Mg$^{2+}$ revealing a dissociative exchange mechanism. Transition interface sampling yields exchange rates within one order of magnitude of experiment, representing a substantial improvement over classical dissociative force fields. In contrast, the NNP-derived solvation free energy significantly underestimates the experimental value, revealing a limitation of the present local NNP architectures for describing ion solvation thermodynamics. Our results demonstrate that DFT-trained NNPs can accurately describe Mg$^{2+}$ hydration structure, diffusion, ion pairing, and exchange kinetics, while highlighting the need for explicit long-range electrostatic treatments to achieve quantitative agreement with experimental ion solvation free energies.

We present the first systematic and comprehensive scan of two-flavon Froggatt-Nielsen (FN) models, employing artificial intelligence techniques to explore the high-dimensional, mixed discrete-continuous parameter space. Extending the standard single-flavon FN framework to a two-flavon setup in which separate flavon fields couple independently to the up- and down-type sectors, we demonstrate that the relative phase between their vacuum expectation values (vevs) provides a natural and generic source of CP violation absent in single-flavon models. To explore this enlarged model space, we cast the search for phenomenologically viable models as a multi-objective optimisation problem, formulating each experimental constraint as a separate objective, and employ the Non-dominated Sorting Genetic Algorithm III to simultaneously fit all 18 FN charges, 45 Wilson coefficients, and flavon parameters to both the quark and lepton sectors. Our approach requires no separate training phase and identifies phenomenologically viable models orders of magnitude faster than prior reinforcement learning methods. Imposing experimental constraints on CKM and PMNS mixing angles and CP phases, charged fermion masses, and neutrino squared-mass differences, we discover over $100\,000$ unique viable models with a remarkably low duplication rate, indicating that the space of valid two-flavon FN realisations has not been exhausted. Both Normal and Inverted neutrino mass squared orderings are realised, with the relative hierarchy between the flavon vevs producing qualitatively distinct predictions for the effective neutrinoless double beta decay mass $m_{ee}$. We further demonstrate the existence of minimal FN realisations with maximal flavon exponent as small as three, and of models reproducing charged fermion masses to within $6\%$ without any dedicated continuous parameter optimisation.

For quantum algorithms to achieve their full potential, we need methodologies to optimize them, such as reaching a given output accuracy with minimal resource costs. Here, we develop such a methodology for a class of Noisy Intermediate-Scale Quantum (NISQ) algorithms. We leverage simulations of a Variational Quantum Eigensolver (VQE) to propose a phenomenological model of such algorithms that captures the complex relationship between algorithmic accuracy, algorithmic resource costs, and the noise that exists in realistic quantum hardware. For this, we take the algorithmic resource cost to be the total number of quantum gate-operations in the algorithm; minimizing this cost typically makes the algorithm faster and more energy-efficient. We consider the subtle trade-off between quantum circuit size (small circuits are too imprecise, but large ones are too noisy), and the number of iterations of that quantum circuit for the full algorithm to sufficiently converge. Using a noise-metric-resource methodology, we identify the sweet spot (of circuit size versus iterations) that minimizes the algorithmic resource costs for a desired algorithm accuracy. It also gives the circuit size that maximizes algorithm accuracy for a fixed resource cost. Our methodology provides a practical guideline for near-term deployment of variational algorithms on realistic noisy hardware, including hardware that uses error mitigation.

We use density functional theory and model Hamiltonians to reveal large spin splitting of bands localized at low-symmetry, ferromagnetic surfaces of bulk antiferromagnets (AFMs). There is great interest in finding new material platforms combining the robustness and ultrafast dynamics of AFMs with large, functional spin splitting which is often restricted to ferromagnets. Here, we show that a subset of AFM surfaces which have symmetry-allowed magnetization can host large spin splitting via bulk degeneracy lifting of sublattice-resolved exchange splittings. Using model Hamiltonians, we show that the spin splitting is maximized for two ferromagnetic surface motifs: terminations with single uncompensated magnetic sublattices, and two-sublattice surfaces whose sublattices are magnetically and electronically compensated in the bulk, but acquire distinct crystal field environments via surface truncation. The latter case can yield FM-like spin splitting magnitudes while also having vanishingly small uncompensated magnetization. In contrast, when surface magnetization arises from relativistic canting on symmetry-connected sublattices, the spin splitting is expected to be small. We confirm these predictions with first-principles calculations of $\mathrm{Cr_2O_3}$ and $\mathrm{FeF_2}$, finding splittings from $\sim10\mathrm{meV}$-$\sim1\mathrm{eV}$ depending on the surface in question. Our findings point to intrinsic surface symmetry breaking as a route to large, functional spin splitting in an expanded range of AFM materials.

Phenomenological model building has traditionally relied on human reasoning: equations are proposed from theoretical intuition, analogy, or empirical convenience, and only then tested against data. Here we show that this cycle can be recast as an iterative AI reasoning process for dynamical dark energy. Our framework uses a large language model to propose equations of state together with cosmological rationales, grounded by retrieval from the dark-energy literature and refined through autonomous evaluation. Each candidate is embedded in a cosmological model, optimized against observations, and assessed using likelihood performance and theoretical consistency. An independent language-model critic scores the physical motivation, novelty, clarity, stability and implementation validity of both the equation and its rationale, allowing subsequent proposals to evolve jointly in mathematical structure and physical reasoning. Applied to cosmological data combinations including supernovae, baryon acoustic oscillations and Planck likelihoods, the framework identifies two parameterizations that, to the best of our knowledge, have not previously been explored and that are competitive with established forms. For Pantheon+ supernovae, DESI DR2 baryon acoustic oscillations and the full Planck 2018 temperature, polarization, and lensing likelihoods, the best AI-selected model attains larger Bayesian evidence than the traditional parameterizations considered here by more than one unit. These results show that AI-guided reasoning can complement physical model building by proposing and evaluating interpretable phenomenological parameterizations for dynamical dark energy.

Efficiently solving the Fokker-Planck equation (FPE) is central to analyzing complex parameterized stochastic systems. However, current numerical methods lack parallel computation capabilities across varying conditions, severely limiting comprehensive parameter exploration and transient analysis. This paper introduces a deep learning-based pseudo-analytical probability solution (PAPS) that, via a single training process, simultaneously resolves transient FPE solutions for arbitrary multi-modal initial distributions, system parameters, and time points. The core idea is to unify initial, transient, and stationary distributions via Gaussian mixture distributions (GMDs) and develop a constraint-preserving autoencoder that bijectively maps constrained GMD parameters to unconstrained, low-dimensional latent representations. In this representation space, the panoramic transient dynamics across varying initial conditions and system parameters can be modeled by a single evolution network. Extensive experiments on paradigmatic systems demonstrate that the proposed PAPS maintains high accuracy while achieving inference speeds four orders of magnitude faster than GPU-accelerated Monte Carlo simulations. This efficiency leap enables previously intractable real-time parameter sweeps and systematic investigations of stochastic bifurcations. By decoupling representation learning from physics-informed transient dynamics, our work establishes a scalable paradigm for probabilistic modeling of multi-dimensional, parameterized stochastic systems.

Valley splitting in strained Si/SiGe quantum wells is a central parameter for silicon spin qubits and is commonly described with envelope-function and effective-mass theories. These models provide a computationally efficient continuum description and have been shown to agree well with atomistic approaches when the confinement potential is slowly varying on the lattice scale. In modern Si/SiGe heterostructures with atomically sharp interfaces and engineered Ge concentration profiles, however, the slowly varying potential approximation underlying conventional (local) envelope-function theory is challenged. We formulate an exact multi-valley envelope-function model by combining Burt-Foreman-type envelope-function theory, which does not rely on the assumption of a slowly varying potential, with a valley-sector decomposition of the Brillouin zone. This construction enforces band-limited envelopes, which satisfy a set of coupled integro-differential equations with a non-local potential energy operator. Using degenerate perturbation theory, we derive the intervalley coupling matrix element within this non-local model and prove that it is strictly invariant under global shifts of the confinement potential (choice of reference energy). We then show that the conventional local envelope model generically violates this invariance due to spectral leakage between valley sectors, leading to an unphysical energy-reference dependence of the intervalley coupling. The resulting ambiguity is quantified by numerical simulations of various engineered Si/SiGe heterostructures. Finally, we propose a simple spectrally filtered local approximation that restores the energy-reference invariance exactly and provides a good approximation to the exact non-local theory.

Brittle materials subjected to thermal shocks experience strong temperature gradients that in turn give rise to mechanical stresses that can be large enough to induce fracture. This work presents a complete model for phase-field fracture for coupled thermo-mechanical problems, wherein the bulk material properties, the material strength, and the fracture toughness are specified independently. The capabilities of the model are assessed across a wide span of scenarios in thermo-mechanical fracture, from the propagation of large pre-existing cracks to crack nucleation under spatially uniform states of stress. In particular, we revisit the controlled quenching of glass plates, and demonstrate how the model captures experimentally observed crack patterns across a range of thermal loads. Ceramic disks subjected to infrared radiation are also examined, with the model reproducing both straight cracks in notched specimens and branching in intact specimens. Finally, ceramic pellets subjected to rapid power pulses are examined, with the model explaining experimental transitions from intact to fractured pellets and the important role of material strength. The results demonstrate that the complete phase-field model unifies the treatment of distinct fracture scenarios under thermal shock, surpassing classical approaches and enabling more reliable prediction of brittle fracture in extreme environments.

Lax-Wendroff methods combined with discontinuous Galerkin/flux reconstruction spatial discretization provide a high-order, single-stage, quadrature-free method for solving hyperbolic conservation laws. In this work, we introduce automatic differentiation (AD) for performing the Cauchy-Kowalewski procedure used in the element-local time average flux computation step (the predictor step) of Lax-Wendroff methods. The application of AD is similar for methods of any order and does not need positivity corrections during the predictor step. This contrasts with the approximate Lax-Wendroff procedure, which requires different finite difference formulas for different orders of the method and positivity corrections in the predictor step for fluxes that can only be computed on admissible states. The method is Jacobian-free and problem-independent, allowing direct application to any physical flux function. Numerical experiments demonstrate the order and positivity preservation of the method. Additionally, performance comparisons indicate that the wall-clock time of automatic differentiation is always on par with the approximate Lax-Wendroff method.

The rapid scaling of deep neural networks comes at the cost of unsustainable power consumption. While optical neural networks offer an alternative, their capabilities remain constrained by the lack of efficient optical nonlinearities. To address this, we propose an optical neural computing architecture by embedding quantum emitters in inverse-designed nanophotonic structures. Due to their saturability, quantum emitters exhibit exceptionally strong nonlinearity compared with conventional materials. Using physics-aware training, we numerically demonstrate that the proposed architecture can solve complex tasks, including nonlinear classification and reinforcement learning, within all-optical neural networks. To enable fair comparison across different platforms, we introduce a framework that quantitatively links nonlinearity to a network's expressive power. Analysis shows that our quantum activation operates at $\text{nW}/μ\text{m}^2$ intensity, which is seven orders of magnitude below the nonlinearity threshold of conventional optical materials. Looking ahead to large language models, we estimate the nonlinearity-limited optical power, which scales sublinearly with model size. Our results indicate that quantum nanophotonics may provide a route toward sustainable AI inference.

The Richards equation, a nonlinear elliptic parabolic equation, is widely used to model infiltration in porous media. We develop a finite element method for solving the Richards equation by introducing a new bounded auxiliary variable to eliminate unbounded terms in the weak formulation of the method. This formulation is discretized using a semi-implicit scheme and the resulting nonlinear system is solved using Newton's method. Our approach eliminates the need of regularization techniques and offers advantages in handling both dry and fully saturated zones. In the proposed techniques, a non-overlapping Schwarz domain decomposition method is used for modeling infiltration in layered soils. We apply the proposed method to solve the Richards equation using the Havercamp and van Genuchten models for the capillary pressure. Numerical experiments are performed to validate the proposed approach, including tests such as modeling flows in fibrous sheets where the initial medium is totally dry, two cases with fully saturated and dry regions, and an infiltration problem in layered soils. The numerical results demonstrate the stability and accuracy of the proposed numerical method. The numerical solutions remain positive in the presence of totally dry zones. The numerical investigations clearly demonstrated the capability of the proposed method to effectively predict the dynamics of flows in unsaturated soils.

In this paper, we derive a generalized second fluctuation-dissipation theorem (FDT) for stochastic dynamical systems in the steady state. The established theory is built upon the Mori-type generalized Langevin equation for stochastic dynamical systems and only uses the properties of the Kolmogorov operator. The new second FDT expresses the memory kernel of the generalized Langevin equation as the correlation function of the fluctuation force plus an additional term. In particular, we show that for nonequilibrium states such as heat transport between two thermostats with different temperatures, the classical second FDT is valid even when the exact form of the steady state distribution is unknown. The obtained theoretical results enable us to construct a data-driven nanoscale fluctuating heat conduction model based on the second FDT. We numerically verify that the new model of heat transfer yields better predictions than the Green-Kubo formula for systems far from the equilibrium.