A factorized latent-conditioning formulation is introduced that jointly learns a neural operator and a low-dimensional latent representation through factorized prediction, trajectory-decoupled sampling, and dimension selection that enables generalization to previously unseen system instances.
Abstract
Neural operators provide data-driven mappings for modeling dynamical systems. Extending them to families of systems typically requires explicit conditioning variables such as physical parameters, geometries, or boundary conditions. In many real-world settings, these quantities are unobserved. Here, we formulate neural operator discovery (NOD) as the problem of learning both shared solution operators and system-specific variation directly from heterogeneous trajectories without access to labeled governing factors. We introduce a factorized latent-conditioning formulation that jointly learns a neural operator and a low-dimensional latent representation through factorized prediction, trajectory-decoupled sampling, and dimension selection. Across diverse systems, the learned latent representation captures the intrinsic dimensionality of system variation and organizes system instances in a smooth and approximately invertible latent structure aligned with the underlying governing factors. This organization enables generalization to previously unseen system instances, including zero-shot extrapolation across regimes and stable long-horizon prediction. These results establish an interpretable paradigm for operator learning in the absence of explicit factor supervision.
Neural PDE solvers provide efficient surrogates for time-dependent physical systems, but autoregressive prediction over long horizons remains challenging because local errors can induce distribution shift and accumulate under recursive deployment. We develop a variational approach to this problem by introducing latent Markov dynamics in which physical states are represented by latent distributions and evolved through probabilistic transitions. The framework is formulated directly on function spaces and specialized to functional Gaussian models, where structured latent perturbations induce a spectral geometry and variational transition alignment regularizes the learned dynamics. We further analyze how these mechanisms affect autoregressive error propagation, providing a theoretical connection between variational training and long-horizon prediction. We instantiate the framework as the Variational Autoencoding Markov Operator (VAMO), which combines spatially resolved latent fields, structured Gaussian perturbations, and a neural-operator transition. Empirically, we demonstrate the effectiveness of VAMO on several fluid-dynamics benchmarks with prediction horizons extending substantially beyond those represented during training, where it consistently reduces error accumulation and improves rollout stability over several deterministic and noise-injection baselines. Overall, these results highlight variational modeling as a complementary approach to robust long-horizon neural PDE dynamics.
Jun-Yi Liao, J. Guilleminot, Vahid Tarokh· 0 citations
Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing partial differential equations (PDEs). Recently, deep learning, as represented by neural operators, has provided a data-driven paradigm for addressing such tasks. However, most existing global neural operators for PDEs require large training datasets and many learnable parameters, with limited interpretability and generalization. We propose the local gradient neural operator (LGNO) as a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems. The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels that resemble discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, rendering the learned operators more transparent. For problems with symmetries, network folding shares equivalent components and reduces parameter counts. We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases. Results show that LGNO maintains accuracy, parameter efficiency, and rollout stability across these tasks, and further exhibits wide applicability to mechanical problems including diffusion, flow, and quantum phenomena.
Bai-Ming Zhang, Jin-Song Tang, Ying Xu et al.· 0 citations
Neural encoding, decoding, and representation-learning approaches have substantially advanced our ability to predict sensory, behavioral, and cognitive variables from neural population activity. At the same time, dynamical systems approaches increasingly model neural computation as the evolution of latent population states governed by recurrent interactions and structured state transitions. Although these frameworks are often presented as competing paradigms, both can successfully reproduce neural observations while still failing to uniquely identify the computational mechanisms implemented by biological circuits. This perspective argues that this reflects a central unresolved challenge in systems neuroscience: observational recordings alone often provide insufficient constraints for distinguishing mechanistically valid neural dynamics from observationally equivalent alternatives. Accordingly, this perspective proposes a unifying framework integrating representational models, latent neural dynamics, identifiability analysis, and perturbation-based validation within a common mechanistic perspective. First, neural representations are discussed as potentially emerging from temporally localized projections of underlying latent dynamical processes evolving on low-dimensional manifolds. Second, recent advances in learning neural evolution operators using recurrent neural networks, latent state-space models, and dynamical system reconstruction methods are reviewed. Third, it is argued that latent trajectories and predictive performance alone do not guarantee mechanistic validity because multiple latent organizations and evolution operators may remain observationally equivalent despite implying distinct causal mechanisms. Finally, perturbation, intervention, and closed-loop neural interfaces are discussed as additional causal constraints capable of falsifying candidate dynamical explanations under targeted manipulation. Across these four principles, the central challenge in modern neuroscience is framed not simply as decoding neural activity or reconstructing latent trajectories, but as determining which inferred dynamical operators remain predictive under intervention and how perturbation can reduce the admissible class of observationally equivalent candidate mechanisms. From this perspective, evolution operators become experimentally testable hypotheses rather than purely descriptive latent models. Integrating latent dynamical modeling with perturbation-based validation may therefore support a transition from prediction-oriented neuroscience toward perturbation-constrained mechanistic dynamical neuroscience.
Armin Hakkak Moghadam Torbati· Journal of Neural Engineerin...· 0 citations
Forecasting the long-horizon evolution of mechanical systems from position-only observations is a pivotal yet difficult task, as hidden velocities and trajectory-specific physical properties must be inferred simultaneously. Although physics-guided neural networks like Lagrangian Neural Networks (LNNs) guarantee physical plausibility, they generally require complete state inputs and lack adaptability to changing system parameters. To break these limitations, we introduce History-informed Lagrangian Neural Networks (HiLNN). Grounded in the insight that temporal position sequences implicitly encode underlying dynamics, HiLNN employs a recurrent encoder to extract a latent context from history. This context not only reconstructs the unobserved initial velocity but also adaptively modulates the mass matrix, potential energy, and damping coefficients of a structured Lagrangian system. By leveraging a differentiable RK4 rollout scheme, the entire pipeline is optimized end-to-end under multi-step trajectory supervision and energy-consistency regularization. Empirical evaluations across conservative, dissipative, and heterogeneous variable-parameter systems show that HiLNN delivers superior long-term prediction accuracy and maintains precise energy profiles compared to state-of-the-art baselines. The source code is publicly available at https://github.com/yingtian22/History-informed-LNN.
Tian-Shuo Zhang, Xianglei Xing, Wenzhe Zhai et al.· 0 citations
Learning solution operators across broad parameter ranges can require substantial coverage of both input functions and physical parameters, particularly for purely data-driven parametric models. In addition, the resulting models may fail silently outside the training distribution. We introduce equation recast, which reformulates parametric operator learning as the learning of a single canonical operator. Parameter-induced operator variations are derived analytically from the governing equation and absorbed into effective sources, enabling zero-shot prediction across new parameter regimes. Across multi-parameter, nonlinear, and singular PDE settings, equation recast supports extrapolation, integrates sparse heterogeneous datasets in a shared canonical representation, and uses loss of convergence as an internal warning signal for failure of the recast iteration. In high-fidelity tokamak simulations for nuclear fusion, the framework unifies electron-temperature data across four device geometries through canonical-domain mapping within one jointly trained operator. Equation recast provides a route toward reusable neural PDE solvers combining equation-guided transfer, data efficiency, and monitorable inference.
Qi-Yun Cheng, Valentin Duruisseaux, C. Clauser et al.· 0 citations
This work presents a population-level inference framework that recovers latent stochastic dynamics directly from snapshot probability distributions by decomposing the observed evolution into an intrinsic latent stochastic process and a discrepancy transport map that captures geometric deformation between the latent and observed probability spaces.
Cheng-Yang Huang, K. Garikipati· arXiv.org· 0 citations
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