Pub Date : 2026-08-31DOI: 10.1016/j.cnsns.2026.110756
Li Xie
We consider a fully parabolic chemotaxis-type system with singular sensitivity and mixed signal production, which is closely related to the urban crime model of Short et al. and the chemotaxis model of Othmer–Stevens. Compared to the crime model, our system lacks a key damping term and thus loses some important decay properties which play a pivotal role in establishing the global solvability of the crime model; However, the signal in our system undergoes diffusion, which may suppress the finite-time blowup of the Othmer–Stevens model, a counterpart in which the signal lacks diffusion. By constructing two different energy functionals, we establish the global classical solvability in the one-dimensional and two-dimensional settings under suitably small initial data.
{"title":"Global smooth solutions in a chemotaxis-type system with singular sensitivity and mixed signal production","authors":"Li Xie","doi":"10.1016/j.cnsns.2026.110756","DOIUrl":"https://doi.org/10.1016/j.cnsns.2026.110756","url":null,"abstract":"We consider a fully parabolic chemotaxis-type system with singular sensitivity and mixed signal production, which is closely related to the urban crime model of Short et al. and the chemotaxis model of Othmer–Stevens. Compared to the crime model, our system lacks a key damping term and thus loses some important decay properties which play a pivotal role in establishing the global solvability of the crime model; However, the signal in our system undergoes diffusion, which may suppress the finite-time blowup of the Othmer–Stevens model, a counterpart in which the signal lacks diffusion. By constructing two different energy functionals, we establish the global classical solvability in the one-dimensional and two-dimensional settings under suitably small initial data.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"11 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2026-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148884154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-31DOI: 10.1016/j.cnsns.2026.110734
Pengxiang Sun, Chen Li, Dachuang Cheng, Penghua Mu
Photonic reservoir computing (PRC) has emerged as a compelling paradigm for next-generation, low-power, and ultra-high-bandwidth neuromorphic computing; however, its on-chip integration faces a critical bottleneck. Traditional time-delay reservoir computing (TDRC) architectures rely heavily on long physical feedback loops that impose a prohibitive footprint, whereas existing feedback-free reservoir computing (FFRC) alternatives suffer from an intrinsic physical limitation of severely depleted memory capacity (MC). To address this challenge, this paper proposes and systematically validates a novel feedback-free photonic reservoir computing (QC-RC) architecture synergistically driven by dual-core nanolasers and quasi-convolutional (QC) encoding. At the physical layer, the architecture leverages the cavity quantum electrodynamics (cQED) effects within on-chip, strongly mutually coupled dual-entity microcavities to reshape the nonlinear transient response dynamics at the microscopic scale. At the algorithmic layer, deep historical correlations are established at the data source through time-domain sliding-window weight superposition without introducing hardware overhead. Utilizing modified nanolaser rate equations that incorporate the Purcell effect alongside multi-dimensional parameter space optimization, our findings demonstrate that at the optimal operating point along the critical steady-state frontier-under standard nanocavity fabrication benchmarks-the long-tail decay of the optical field in the physical microcavity achieves a highly compatible mechanistic synergy with the pure algorithm-domain QC encoding. Furthermore, we comprehensively investigate the parameter boundary effects and the evolutionary diversity of the state space within this system, successfully elevating the total MC to a high-level plateau of 28.0. In standard benchmark tasks, the proposed system yields a normalized mean square error (NMSE) as low as 0.0035 for chaotic time-series prediction and achieves an exceptionally low symbol error rate (SER) in channel equalization. This work substantially decouples the system memory depth from its physical hardware dimensions, thereby offering a highly resilient theoretical and technical framework with favorable engineering tolerance for future ultra-compact, high-throughput, on-chip all-optical intelligent computing.
{"title":"Feedback-free photonic reservoir computing with high memory capacity driven by mutually coupled dual-core nanolasers and quasi-convolutional encoding","authors":"Pengxiang Sun, Chen Li, Dachuang Cheng, Penghua Mu","doi":"10.1016/j.cnsns.2026.110734","DOIUrl":"https://doi.org/10.1016/j.cnsns.2026.110734","url":null,"abstract":"Photonic reservoir computing (PRC) has emerged as a compelling paradigm for next-generation, low-power, and ultra-high-bandwidth neuromorphic computing; however, its on-chip integration faces a critical bottleneck. Traditional time-delay reservoir computing (TDRC) architectures rely heavily on long physical feedback loops that impose a prohibitive footprint, whereas existing feedback-free reservoir computing (FFRC) alternatives suffer from an intrinsic physical limitation of severely depleted memory capacity (MC). To address this challenge, this paper proposes and systematically validates a novel feedback-free photonic reservoir computing (QC-RC) architecture synergistically driven by dual-core nanolasers and quasi-convolutional (QC) encoding. At the physical layer, the architecture leverages the cavity quantum electrodynamics (cQED) effects within on-chip, strongly mutually coupled dual-entity microcavities to reshape the nonlinear transient response dynamics at the microscopic scale. At the algorithmic layer, deep historical correlations are established at the data source through time-domain sliding-window weight superposition without introducing hardware overhead. Utilizing modified nanolaser rate equations that incorporate the Purcell effect alongside multi-dimensional parameter space optimization, our findings demonstrate that at the optimal operating point along the critical steady-state frontier-under standard nanocavity fabrication benchmarks-the long-tail decay of the optical field in the physical microcavity achieves a highly compatible mechanistic synergy with the pure algorithm-domain QC encoding. Furthermore, we comprehensively investigate the parameter boundary effects and the evolutionary diversity of the state space within this system, successfully elevating the total MC to a high-level plateau of 28.0. In standard benchmark tasks, the proposed system yields a normalized mean square error (NMSE) as low as 0.0035 for chaotic time-series prediction and achieves an exceptionally low symbol error rate (SER) in channel equalization. This work substantially decouples the system memory depth from its physical hardware dimensions, thereby offering a highly resilient theoretical and technical framework with favorable engineering tolerance for future ultra-compact, high-throughput, on-chip all-optical intelligent computing.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"51 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2026-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148884180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-31DOI: 10.1016/j.cnsns.2026.110747
Yujie Wang, Zhen Wang, Junyu Chen
In this paper, we consider the numerical solution and parameter identification of the nonlinear time-fractional mobile/immobile equation, which is often used to describe anomalous solute transport in heterogeneous media. We propose an efficient fully discrete numerical scheme, in which the local discontinuous Galerkin method is employed in space, the averaged L1 approximation on graded meshes is adopted for the time-fractional derivative, and the first-order backward Euler method is used for the first-order time derivative. For solutions with weak regularity at the initial time, we prove the stability of the scheme in the spatial H1-norm and establish an optimal error estimate. Based on the good performance of the forward solver, we design a parameter inversion procedure that utilizes the particle swarm optimization algorithm to simultaneously recover the fractional order and the transport coefficients from observation data. Extensive numerical experiments are carried out, including synthetic examples corrupted by white noise with different intensities and a classical mountain stream tracer test. Numerical results demonstrate that the proposed framework offers high accuracy and practical value for both forward simulation and inverse modeling of time-fractional diffusion processes.
{"title":"A Local Discontinuous Galerkin Method for the Time-Fractional MIM Equation: Spatial H1-Norm Analysis and Parameter Identification","authors":"Yujie Wang, Zhen Wang, Junyu Chen","doi":"10.1016/j.cnsns.2026.110747","DOIUrl":"https://doi.org/10.1016/j.cnsns.2026.110747","url":null,"abstract":"In this paper, we consider the numerical solution and parameter identification of the nonlinear time-fractional mobile/immobile equation, which is often used to describe anomalous solute transport in heterogeneous media. We propose an efficient fully discrete numerical scheme, in which the local discontinuous Galerkin method is employed in space, the averaged L1 approximation on graded meshes is adopted for the time-fractional derivative, and the first-order backward Euler method is used for the first-order time derivative. For solutions with weak regularity at the initial time, we prove the stability of the scheme in the spatial <ce:italic>H</ce:italic><ce:sup loc=\"post\">1</ce:sup>-norm and establish an optimal error estimate. Based on the good performance of the forward solver, we design a parameter inversion procedure that utilizes the particle swarm optimization algorithm to simultaneously recover the fractional order and the transport coefficients from observation data. Extensive numerical experiments are carried out, including synthetic examples corrupted by white noise with different intensities and a classical mountain stream tracer test. Numerical results demonstrate that the proposed framework offers high accuracy and practical value for both forward simulation and inverse modeling of time-fractional diffusion processes.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"3 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2026-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148884155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-29DOI: 10.1016/j.cnsns.2026.110732
Sijing Lai, Wenxuan Xie, Yibao Li
In this paper, we propose a fully physics-informed and microstructurally interconnected multi-scale topology optimization method within the phase-field framework. The proposed approach establishes a bi-scale coupled alternating optimization architecture to achieve a fully physics-driven multi-scale design through the co-evolution of macro-micro coupled displacement neural networks and macro-micro coupled phase-field neural networks. By integrating microstructural effects into the macro-scale response via homogenization theory, a multi-scale coupled energy functional is constructed. Specifically, the physical loss of the coupled macro-micro displacement neural networks is formulated according to the principle of minimum potential energy and homogenization theory. Simultaneously, a multi-scale phase-field energy functional is introduced within the phase-field framework, where a connectivity-index penalty term is incorporated into the loss function of the coupled macro-micro phase-field neural networks. This formulation incorporates the coupled multi-scale physical relations into the network training process. Furthermore, macroscopic single-scale pre-optimization is performed to initialize the subsequent multi-scale optimization. Automatic differentiation is employed to circumvent the complex sensitivity analysis process. Various numerical experiments demonstrate the validity and effectiveness of the proposed method.
{"title":"A fully physics-informed and microstructurally interconnected multi-scale topology optimization method within the phase-field framework","authors":"Sijing Lai, Wenxuan Xie, Yibao Li","doi":"10.1016/j.cnsns.2026.110732","DOIUrl":"https://doi.org/10.1016/j.cnsns.2026.110732","url":null,"abstract":"In this paper, we propose a fully physics-informed and microstructurally interconnected multi-scale topology optimization method within the phase-field framework. The proposed approach establishes a bi-scale coupled alternating optimization architecture to achieve a fully physics-driven multi-scale design through the co-evolution of macro-micro coupled displacement neural networks and macro-micro coupled phase-field neural networks. By integrating microstructural effects into the macro-scale response via homogenization theory, a multi-scale coupled energy functional is constructed. Specifically, the physical loss of the coupled macro-micro displacement neural networks is formulated according to the principle of minimum potential energy and homogenization theory. Simultaneously, a multi-scale phase-field energy functional is introduced within the phase-field framework, where a connectivity-index penalty term is incorporated into the loss function of the coupled macro-micro phase-field neural networks. This formulation incorporates the coupled multi-scale physical relations into the network training process. Furthermore, macroscopic single-scale pre-optimization is performed to initialize the subsequent multi-scale optimization. Automatic differentiation is employed to circumvent the complex sensitivity analysis process. Various numerical experiments demonstrate the validity and effectiveness of the proposed method.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"24 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2026-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148884181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-29DOI: 10.1016/j.cnsns.2026.110748
Łukasz Płociniczak, Marek A. Teuerle, Hubert Woszczek
We analyze a class of linear partial differential equations that arise as deterministic descriptions of the scaling limits of Lévy walks, in which transport is driven by a convex combination of fractional material derivatives and a source term. Using techniques of Fourier-Laplace transforms, we first prove the existence of mild solutions for continuous initial data. Using a recently obtained pointwise representation of the fractional material derivative, we then identify a necessary and sufficient condition on the source term that guaranties the solution to remain a probability density for all times (non-negativity and unit mass). Motivated by the need to preserve these probabilistic properties in computations, we construct a finite-volume discretization that is probability conservative by construction. We establish discrete stability and a convergence result for the continuous weak solution as space and time steps tend to zero. Extensive numerical experiments validate the scheme: total mass is conserved, non-negativity is maintained, and the computed solutions reproduce the known analytic representations of the probability density functions associated with the Lévy walk process. The combined theoretical and numerical framework provides a reliable tool for studying anomalous transport governed by fractional dynamics.
{"title":"Convex combination of fractional material derivatives governing the scaling limits of asymmetric Lévy walks","authors":"Łukasz Płociniczak, Marek A. Teuerle, Hubert Woszczek","doi":"10.1016/j.cnsns.2026.110748","DOIUrl":"https://doi.org/10.1016/j.cnsns.2026.110748","url":null,"abstract":"We analyze a class of linear partial differential equations that arise as deterministic descriptions of the scaling limits of Lévy walks, in which transport is driven by a convex combination of fractional material derivatives and a source term. Using techniques of Fourier-Laplace transforms, we first prove the existence of mild solutions for continuous initial data. Using a recently obtained pointwise representation of the fractional material derivative, we then identify a necessary and sufficient condition on the source term that guaranties the solution to remain a probability density for all times (non-negativity and unit mass). Motivated by the need to preserve these probabilistic properties in computations, we construct a finite-volume discretization that is probability conservative by construction. We establish discrete stability and a convergence result for the continuous weak solution as space and time steps tend to zero. Extensive numerical experiments validate the scheme: total mass is conserved, non-negativity is maintained, and the computed solutions reproduce the known analytic representations of the probability density functions associated with the Lévy walk process. The combined theoretical and numerical framework provides a reliable tool for studying anomalous transport governed by fractional dynamics.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"43 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2026-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148884156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-28DOI: 10.1016/j.cnsns.2026.110743
Zihao Shi, Dongling Wang
Physics-informed neural networks and related space–time neural solvers often treat time as an additional input variable, which can obscure temporal causality during approximation and optimization. Neural Galerkin methods address this issue by evolving the neural approximation sequentially in time on a parametric solution manifold. However, existing time-evolving natural Galerkin (TENG) formulations do not, in general, preserve the invariants of conservative PDEs: the time-discrete target state may already violate an invariant, and the subsequent parameter update may drift further from the corresponding invariant manifold. We therefore develop a relaxation–projection enhancement of TENG, termed RP–TENG. The first ingredient is a relaxation Runge–Kutta target construction that enforces a prescribed discrete invariant while retaining the formal order of the underlying explicit integrator. The second ingredient is a constraint-aware parameter update followed by a projection step that returns the network parameters to the discrete invariant manifold. The resulting method remains mesh-free and advances the dynamics directly in parameter space, without solving nonlinear algebraic systems arising from a structure-preserving spatial discretization. Numerical experiments for the inviscid Burgers equation, the Korteweg–de Vries equation, the acoustic wave equation, and a shallow-water model show that RP–TENG substantially improves invariant preservation and, in the tested regimes, often improves the accuracy and robustness of the computed trajectories relative to the baseline TENG method.
{"title":"Preserving conservation laws in the time-evolving natural Galerkin method via relaxation and projection","authors":"Zihao Shi, Dongling Wang","doi":"10.1016/j.cnsns.2026.110743","DOIUrl":"https://doi.org/10.1016/j.cnsns.2026.110743","url":null,"abstract":"Physics-informed neural networks and related space–time neural solvers often treat time as an additional input variable, which can obscure temporal causality during approximation and optimization. Neural Galerkin methods address this issue by evolving the neural approximation sequentially in time on a parametric solution manifold. However, existing time-evolving natural Galerkin (TENG) formulations do not, in general, preserve the invariants of conservative PDEs: the time-discrete target state may already violate an invariant, and the subsequent parameter update may drift further from the corresponding invariant manifold. We therefore develop a relaxation–projection enhancement of TENG, termed RP–TENG. The first ingredient is a relaxation Runge–Kutta target construction that enforces a prescribed discrete invariant while retaining the formal order of the underlying explicit integrator. The second ingredient is a constraint-aware parameter update followed by a projection step that returns the network parameters to the discrete invariant manifold. The resulting method remains mesh-free and advances the dynamics directly in parameter space, without solving nonlinear algebraic systems arising from a structure-preserving spatial discretization. Numerical experiments for the inviscid Burgers equation, the Korteweg–de Vries equation, the acoustic wave equation, and a shallow-water model show that RP–TENG substantially improves invariant preservation and, in the tested regimes, often improves the accuracy and robustness of the computed trajectories relative to the baseline TENG method.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"43 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2026-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148884182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-27DOI: 10.1016/j.cnsns.2026.110727
Taotao Zhao, Zhijian Ji, Lanhao Zhao, Linrong Tan
We study stochastic swarm guidance using a fixed-dimensional cue broadcast to all agents. Broadcast herdability requires sufficient terminal target-tube mass while keeping unsafe and restricted masses below prescribed thresholds throughout the horizon. For the associated Fokker–Planck density, a conditional Lyapunov theorem converts a pointwise generator inequality and analytic set-domination constants into simultaneous mass bounds while quantifying the authority visible through the broadcast subspace. We also give a coefficient-based sufficient condition for mixed-potential feedback and a finite-particle implication conditional on set-functional tracking. In the baseline large-ensemble run, the proposed controller achieved terminal target-tube mass 0.9923 and peak unsafe and restricted masses 0 and 0.0019; no sampled threshold violation occurred in 30 independent 700-particle trials. A second, interaction-active geometry met the sampled thresholds, while robustness sweeps showed performance loss as diffusion or interaction strength increased.
{"title":"Broadcast Herdability of Stochastic Swarm Densities","authors":"Taotao Zhao, Zhijian Ji, Lanhao Zhao, Linrong Tan","doi":"10.1016/j.cnsns.2026.110727","DOIUrl":"https://doi.org/10.1016/j.cnsns.2026.110727","url":null,"abstract":"We study stochastic swarm guidance using a fixed-dimensional cue broadcast to all agents. Broadcast herdability requires sufficient terminal target-tube mass while keeping unsafe and restricted masses below prescribed thresholds throughout the horizon. For the associated Fokker–Planck density, a conditional Lyapunov theorem converts a pointwise generator inequality and analytic set-domination constants into simultaneous mass bounds while quantifying the authority visible through the broadcast subspace. We also give a coefficient-based sufficient condition for mixed-potential feedback and a finite-particle implication conditional on set-functional tracking. In the baseline large-ensemble run, the proposed controller achieved terminal target-tube mass 0.9923 and peak unsafe and restricted masses 0 and 0.0019; no sampled threshold violation occurred in 30 independent 700-particle trials. A second, interaction-active geometry met the sampled thresholds, while robustness sweeps showed performance loss as diffusion or interaction strength increased.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"6 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2026-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148884183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The data-driven discovery of nonlinear partial differential equations (PDEs) from sparse and potentially noise-free observations remains a significant challenge in computational physics. Hidden Physics Models (HPMs), which leverage Gaussian Process (GP) priors to encode underlying physical laws, provide a powerful probabilistic framework for this task. However, the application of standard covariance functions, such as the Squared Exponential (SE) kernel, frequently leads to severe ill-conditioning of the Gram matrix when high-order differential operators are involved. This numerical instability creates a highly non-convex and “jagged” likelihood landscape, which thwarts gradient-based optimization and leads to a catastrophic breakdown in parameter estimation.In this paper, we propose an enhanced HPMs framework based on a Hybrid Kernel approach to overcome these fundamental numerical bottlenecks. By constructing a composite covariance structure that integrates an expressive base kernel with a structurally regularizing Inverse Multiquadric (IMQ) component, we effectively bound the condition number of the system. This structural stabilization smooths the Negative Log Marginal Likelihood (NLML) surface, restoring the robustness of quasi-Newton optimizers like L-BFGS in the noise-free limit. The proposed framework is rigorously validated through a series of canonical nonlinear problems, including the Burgers’, Korteweg-de Vries (KdV), Kuramoto-Sivashinsky (KS), and 2D incompressible Navier-Stokes equations. Numerical results demonstrate that the hybrid approach successfully resolves the accuracy-stability trade-off, achieving high-fidelity parameter recovery and precise system identification from extremely sparse data (utilizing less than 0.8% of the available spatiotemporal domain). Our findings suggest that hybrid kernelization is essential for the stable and accurate discovery of complex dynamical systems where standard single-kernel HPMs fail due to numerical singularity.
{"title":"Stabilized Hidden Physics Models via Hybrid Kernels for the Data-Driven Discovery of Nonlinear PDEs","authors":"Meysam Cheraghi, Mohsen Esmaeilbeigi, Ebrahim Nazari","doi":"10.1016/j.cnsns.2026.110733","DOIUrl":"https://doi.org/10.1016/j.cnsns.2026.110733","url":null,"abstract":"The data-driven discovery of nonlinear partial differential equations (PDEs) from sparse and potentially noise-free observations remains a significant challenge in computational physics. Hidden Physics Models (HPMs), which leverage Gaussian Process (GP) priors to encode underlying physical laws, provide a powerful probabilistic framework for this task. However, the application of standard covariance functions, such as the Squared Exponential (SE) kernel, frequently leads to severe ill-conditioning of the Gram matrix when high-order differential operators are involved. This numerical instability creates a highly non-convex and “jagged” likelihood landscape, which thwarts gradient-based optimization and leads to a catastrophic breakdown in parameter estimation.In this paper, we propose an enhanced HPMs framework based on a Hybrid Kernel approach to overcome these fundamental numerical bottlenecks. By constructing a composite covariance structure that integrates an expressive base kernel with a structurally regularizing Inverse Multiquadric (IMQ) component, we effectively bound the condition number of the system. This structural stabilization smooths the Negative Log Marginal Likelihood (NLML) surface, restoring the robustness of quasi-Newton optimizers like L-BFGS in the noise-free limit. The proposed framework is rigorously validated through a series of canonical nonlinear problems, including the Burgers’, Korteweg-de Vries (KdV), Kuramoto-Sivashinsky (KS), and 2D incompressible Navier-Stokes equations. Numerical results demonstrate that the hybrid approach successfully resolves the accuracy-stability trade-off, achieving high-fidelity parameter recovery and precise system identification from extremely sparse data (utilizing less than 0.8% of the available spatiotemporal domain). Our findings suggest that hybrid kernelization is essential for the stable and accurate discovery of complex dynamical systems where standard single-kernel HPMs fail due to numerical singularity.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"13 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2026-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148884184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-26DOI: 10.1016/j.cnsns.2026.110729
Ajaykumar V. Tala, Ankit K. Shah
Sparse identification of nonlinear dynamics (SINDy) discovers governing equations from time-series data, but its Sequentially Thresholded Least-Squares (STLSQ) core depends critically on a sparsity threshold λ that is usually chosen by manual tuning or grid search, limiting reproducibility. We introduce PACE (Pareto–AICc–CV Consensus Estimator), an automated framework that selects the threshold parameter. PACEsweeps a log-spaced grid and computes two methodologically distinct estimates: a Pareto-elbow estimate λP from the residual–sparsity trade-off and an h-block cross-validation (CV) estimate λCV that respects temporal autocorrelation. These define a candidate region within which the corrected Akaike Information Criterion (AICc) selects the final threshold λ*; a CV guard verifies predictive competitiveness, and an AICc well-width diagnostic assigns calibrated confidence labels (High, Moderate, Low).
{"title":"PACE: Pareto–AICc–CV Consensus Estimator for Automated Threshold Selection in Sparse Identification of Nonlinear Dynamical Systems","authors":"Ajaykumar V. Tala, Ankit K. Shah","doi":"10.1016/j.cnsns.2026.110729","DOIUrl":"https://doi.org/10.1016/j.cnsns.2026.110729","url":null,"abstract":"Sparse identification of nonlinear dynamics (SINDy) discovers governing equations from time-series data, but its Sequentially Thresholded Least-Squares (STLSQ) core depends critically on a sparsity threshold <ce:italic>λ</ce:italic> that is usually chosen by manual tuning or grid search, limiting reproducibility. We introduce <ce:small-caps>PACE</ce:small-caps> (Pareto–AICc–CV Consensus Estimator), an automated framework that selects the threshold parameter. <ce:small-caps>PACE</ce:small-caps>sweeps a log-spaced grid and computes two methodologically distinct estimates: a Pareto-elbow estimate <ce:italic>λ<ce:inf loc=\"post\">P</ce:inf></ce:italic> from the residual–sparsity trade-off and an h-block cross-validation (CV) estimate <ce:italic>λ<ce:inf loc=\"post\">CV</ce:inf></ce:italic> that respects temporal autocorrelation. These define a candidate region within which the corrected Akaike Information Criterion (AICc) selects the final threshold <ce:italic>λ</ce:italic>*; a CV guard verifies predictive competitiveness, and an AICc well-width diagnostic assigns calibrated confidence labels (<ce:small-caps>High</ce:small-caps>, <ce:small-caps>Moderate</ce:small-caps>, <ce:small-caps>Low</ce:small-caps>).","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2026-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148853167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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