SpatioTemporal Causal Network Diagnostics for Geographic Tipping Point Early Warning

Geographic tipping points in ecosystems, climate subsystems, or ice sheets pose severe challenges for localized early warning. Classical spatial indicators such as Moran's I summarize global spatial structure, but they struggle with three issues: spatial dilution, Euclidean assumptions, and correlated noise. This paper introduces SpatioTemporal Causal Network Diagnostics (ST-CND), a framework that addresses these three issues by representing the geographic field as a time-evolving directed causal network. The core workflow is as follows: (1) infer which spatial nodes help predict other nodes via transfer entropy, replacing fixed Euclidean neighborhoods with data-driven information-flow topology; (2) estimate local recovery rates within each candidate subnetwork via dynamic mode decomposition; and (3) identify the most vulnerable subnetwork by combining three signals, namely high internal fluctuation, high internal synchronization, and low external coupling, thereby suppressing false alarms from spatially correlated noise. Validated on synthetic bifurcations and two observational sea-surface temperature benchmarks, namely Indo-Pacific SST and North Atlantic AMOC, ST-CND delivers localized and interpretable warnings. On the AMOC task, it achieves an AUROC of 0.783 and a critical-subnetwork IoU of 0.378, outperforming recurrence-network and lambda-AR1 baselines. The framework provides an interpretable and scalable pipeline for spatial early warning in Earth system science.

Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations

We propose HIN-LRI, a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error. A base low-regularity integrator provides a consistent first-order approximation to nonlinear dispersive PDEs, while a lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close. An explicit time-step scaling on the neural correction ensures that its Lipschitz contribution remains $\mathcal{O}(τ)$, yielding a Gronwall stability factor bounded uniformly in the step size and independent of the spatial resolution. The network is trained end-to-end through a solver-in-the-loop objective that unrolls the full iteration and penalises trajectory error in a Bourgain-type norm, aligning learning with multi-step solver dynamics rather than isolated one-step targets. Under stated assumptions, the global error satisfies $C(\varepsilon_{net}+δ)\,τ^γ\ln(1/τ)$, where $\varepsilon_{net}$ measures the network approximation quality and $δ$ the training shortfall. Experiments on three dispersive benchmarks with rough data show that HIN-LRI improves accuracy over analytical integrators, splitting methods, and neural PDE surrogates, with stable spatial refinement, effective out-of-distribution transfer, and modest online overhead.

Categorical Optimization with Bayesian Anchored Latent Trust Regions for Structural Design under High-Dimensional Uncertainty

Categorical structural optimization under aleatoric uncertainty is challenging because each design variable must be selected from a finite catalog of admissible instances, while each candidate design may require expensive stochastic finite-element evaluations. Existing latent-space optimization strategies can reduce the dimensionality of catalog attributes, but they often treat the reduced space as a continuous search domain. The resulting continuous optimum must then be rounded off to a nearby catalog instance, which may alter the objective value, constraint status, or physical interpretation of the design. To address this issue, this paper proposes the \textbf{C}ategorical \textbf{O}ptimization with \textbf{B}ayesian \textbf{A}nchored \textbf{L}atent \textbf{T}rust Regions (\textbf{COBALT}) framework for high-dimensional categorical Optimization Under Uncertainty. COBALT first embeds the physical catalog into a low-dimensional latent representation and locks the mapped instances as a discrete anchored graph. A data-independent random tree decomposition is then used to provide bounded-complexity additive modeling over high-dimensional categorical variables. On this anchored domain, an additive SAAS-GP surrogate is fitted to heteroscedastic MC-FEA observations, and a trust-region discrete graph acquisition search selects the next admissible catalog configuration without continuous relaxation or rounding-off. The proposed strategy is applied to robust design optimization of complex bar structures, considering structural weight, strain energy, and local buckling performance. By evaluating only valid catalog designs through the MC-FEA oracle, COBALT preserves physical admissibility throughout the active learning loop and improves the efficiency of robust categorical structural optimization.

Conflict-Aware Harmonized Rotational Gradient for Multiscale Kinetic Regimes

In this paper, we propose a harmonized rotational gradient method, termed HRGrad, for simultaneously tackling multiscale time-dependent kinetic problems with varying small parameters. These parameters exhibit asymptotic transitions from microscopic to macroscopic physics, making it a challenging multi-task problem to solve over all ranges simultaneously. Solving tasks in different asymptotic regions often encounter gradient conflicts, which can lead to the failure of multi-task learning. To address this challenge, we explicitly encode a hidden representation of these parameters, ensuring that the corresponding solving tasks are serialized for simultaneous training. Furthermore, to mitigate gradient conflicts, we segment the prediction results to construct task losses and introduce a novel gradient alignment metric to ensure a positive dot product between the final update and each loss-specific gradient. This metric maintains consistent optimization rates for all task losses and dynamically adjusts gradient magnitudes based on conflict levels. Moreover, we provide a mathematical proof demonstrating the convergence of the HRGrad method, which is evaluated across a range of challenging asymptotic-preserving neural networks (APNNs) scenarios. We conduct an extensive set of experiments encompassing the Bhatnagar-Gross-Krook (BGK) equation and the linear transport equation in all ranges of Knudsen number. Our results indicate that HRGrad effectively overcomes the `failure modes' of APNNs in these problems.

Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domains

In this paper, we propose a stochastic-dimension frozen sampled neural network (SD-FSNN) for solving a class of high-dimensional Gross-Pitaevskii equations (GPEs) on unbounded domains. SD-FSNN is unbiased across all dimensions, and its computational cost is independent of the dimension, avoiding the exponential growth in computational and memory costs associated with Hermite-basis discretizations. Additionally, we randomly sample the hidden weights and biases of the neural network, significantly outperforming iterative, gradient-based optimization methods in terms of training time and accuracy. Furthermore, we employ a space-time separation strategy, using adaptive ordinary differential equation (ODE) solvers to update the evolution coefficients and incorporate temporal causality. To preserve the structure of the GPEs, we integrate a Gaussian-weighted ansatz into the neural network to enforce exponential decay at infinity, embed a normalization projection layer for mass normalization, and add an energy conservation constraint to mitigate long-time numerical dissipation. Comparative experiments with existing methods demonstrate the superior performance of SD-FSNN across a range of spatial dimensions and interaction parameters. Compared to existing random-feature methods, SD-FSNN reduces the complexity from linear to dimension-independent. Additionally, SD-FSNN achieves better accuracy and faster training compared to general high-dimensional solvers, while focusing specifically on high-dimensional GPEs on unbounded domains.

Stochastic Dimension Implicit Functional Projections for Exact Integral Conservation in High-Dimensional PINNs

Enforcing exact macroscopic conservation laws, such as mass and energy, in neural partial differential equation (PDE) solvers is computationally challenging in high dimensions. Traditional discrete projections rely on deterministic quadrature that scales poorly and restricts mesh-free formulations like PINNs. Furthermore, high-order operators incur heavy memory overhead, and generic optimization often lacks convergence guarantees for non-convex conservation manifolds. To address this, we propose the Stochastic Dimension Implicit Functional Projection (SDIFP) framework. Instead of projecting discrete vectors, SDIFP applies a global affine transformation to the continuous network output. This yields closed-form solutions for integral constraints via detached Monte Carlo (MC) quadrature, bypassing spatial grid dependencies. For scalable training, we introduce a doubly-stochastic unbiased gradient estimator (DS-UGE). By decoupling spatial sampling from differential operator subsampling, the DS-UGE reduces memory complexity from $\mathcal{O}(M \times N_{\mathcal{L}})$ to $\mathcal{O}(N \times |\mathcal{I}|)$. SDIFP mitigates sampling variance, preserves solution regularity, and maintains $\mathcal{O}(1)$ inference efficiency, providing a scalable, mesh-free approach for solving conservative high-dimensional PDEs.

Constitutive parameterized deep energy method for solid mechanics problems with random material parameters

In practical structural design and solid mechanics simulations, material properties inherently exhibit random variations within bounded intervals. However, evaluating mechanical responses under continuous material uncertainty remains a persistent challenge. Traditional numerical approaches, such as the Finite Element Method (FEM), incur prohibitive computational costs as they require repeated mesh discretization and equation solving for every parametric realization. Similarly, data-driven surrogate models depend heavily on massive, high-fidelity datasets, while standard physics-informed frameworks (e.g., the Deep Energy Method) strictly demand complete retraining from scratch whenever material parameters change. To bridge this critical gap, we propose the Constitutive Parameterized Deep Energy Method (CPDEM). In this purely physics-driven framework, the strain energy density functional is reformulated by encoding a latent representation of stochastic constitutive parameters. By embedding material parameters directly into the neural network alongside spatial coordinates, CPDEM transforms conventional spatial collocation points into parameter-aware material points. Trained in an unsupervised manner via expected energy minimization over the parameter domain, the pre-trained model continuously learns the solution manifold. Consequently, it enables zero-shot, real-time inference of displacement fields for unknown material parameters without requiring any dataset generation or model retraining. The proposed method is rigorously validated across diverse benchmarks, including linear elasticity, finite-strain hyperelasticity, and complex highly nonlinear contact mechanics. To the best of our knowledge, CPDEM represents the first purely physics-driven deep learning paradigm capable of simultaneously and efficiently handling continuous multi-parameter variations in solid mechanics.

Stochastic Dimension-Free Zeroth-Order Estimator for High-Dimensional and High-Order PINNs

Physics-Informed Neural Networks (PINNs) for high-dimensional and high-order partial differential equations (PDEs) are primarily constrained by the $\mathcal{O}(d^k)$ spatial derivative complexity and the $\mathcal{O}(P)$ memory overhead of backpropagation (BP). While randomized spatial estimators successfully reduce the spatial complexity to $\mathcal{O}(1)$, their reliance on first-order optimization still leads to prohibitive memory consumption at scale. Zeroth-order (ZO) optimization offers a BP-free alternative; however, naively combining randomized spatial operators with ZO perturbations triggers a variance explosion of $\mathcal{O}(1/\varepsilon^2)$, leading to numerical divergence. To address these challenges, we propose the \textbf{S}tochastic \textbf{D}imension-free \textbf{Z}eroth-order \textbf{E}stimator (\textbf{SDZE}), a unified framework that achieves dimension-independent complexity in both space and memory. Specifically, SDZE leverages \emph{Common Random Numbers Synchronization (CRNS)} to algebraically cancel the $\mathcal{O}(1/\varepsilon^2)$ variance by locking spatial random seeds across perturbations. Furthermore, an \emph{implicit matrix-free subspace projection} is introduced to reduce parameter exploration variance from $\mathcal{O}(P)$ to $\mathcal{O}(r)$ while maintaining an $\mathcal{O}(1)$ optimizer memory footprint. Empirical results demonstrate that SDZE enables the training of 10-million-dimensional PINNs on a single NVIDIA A100 GPU, delivering significant improvements in speed and memory efficiency over state-of-the-art baselines.

Manifold-Orthogonal Dual-spectrum Extrapolation for Parameterized Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have achieved notable success in modeling dynamical systems governed by partial differential equations (PDEs). To avoid computationally expensive retraining under new physical conditions, parameterized PINNs (P$^2$INNs) commonly adapt pre-trained operators using singular value decomposition (SVD) for out-of-distribution (OOD) regimes. However, SVD-based fine-tuning often suffers from rigid subspace locking and truncation of important high-frequency spectral modes, limiting its ability to capture complex physical transitions. While parameter-efficient fine-tuning (PEFT) methods appear to be promising alternatives, applying conventional adapters such as LoRA to P$^2$INNs introduces a severe Pareto trade-off, as additive updates increase parameter overhead and disrupt the structured physical manifolds inherent in operator representations. To address these limitations, we propose Manifold-Orthogonal Dual-spectrum Extrapolation (MODE), a lightweight micro-architecture designed for physics operator adaptation. MODE decomposes physical evolution into complementary mechanisms including principal-spectrum dense mixing that enables cross-modal energy transfer within frozen orthogonal bases, residual-spectrum awakening that activates high-frequency spectral components through a single trainable scalar, and affine Galilean unlocking that explicitly isolates spatial translation dynamics. Experiments on challenging PDE benchmarks including the 1D Convection--Diffusion--Reaction equation and the 2D Helmholtz equation demonstrate that MODE achieves strong out-of-distribution generalization while preserving the minimal parameter complexity of native SVD and outperforming existing PEFT-based baselines.

Disentangled Latent Dynamics Manifold Fusion for Solving Parameterized PDEs

Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable. The problem becomes even more severe when the model must also predict beyond the training time range. Existing methods usually cannot handle parameter generalization and temporal extrapolation at the same time. Standard parameterized models treat time as just another input and therefore fail to capture intrinsic dynamics, while recent continuous-time latent methods often rely on expensive test-time auto-decoding for each instance, which is inefficient and can disrupt continuity across the parameterized solution space. To address this, we propose Disentangled Latent Dynamics Manifold Fusion (DLDMF), a physics-informed framework that explicitly separates space, time, and parameters. Instead of unstable auto-decoding, DLDMF maps PDE parameters directly to a continuous latent embedding through a feed-forward network. This embedding initializes and conditions a latent state whose evolution is governed by a parameter-conditioned Neural ODE. We further introduce a dynamic manifold fusion mechanism that uses a shared decoder to combine spatial coordinates, parameter embeddings, and time-evolving latent states to reconstruct the corresponding spatiotemporal solution. By modeling prediction as latent dynamic evolution rather than static coordinate fitting, DLDMF reduces interference between parameter variation and temporal evolution while preserving a smooth and coherent solution manifold. As a result, it performs well on unseen parameter settings and in long-term temporal extrapolation. Experiments on several benchmark problems show that DLDMF consistently outperforms state-of-the-art baselines in accuracy, parameter generalization, and extrapolation robustness.