time-derivative space construction

Neural operators employed as surrogates in PDE-driven outer-loop tasks—such as PDE-constrained optimization and Bayesian inverse problems—suffer from poor derivative sensitivity, hindering gradient-based algorithms. Method: We analyze the joint approximation error of nonlinear operators and their Fréchet derivatives in separable infinite-dimensional Hilbert spaces under Sobolev norms and Gaussian input measures. We propose Derivative-Informed Subspace (DIS) reduction—a novel dimensionality reduction technique—and establish, for the first time, a rigorous error decomposition theory linking DIS to principal component analysis (PCA) within reduced-basis neural operators (RBNOs). Contribution/Results: We prove that DIS bases strictly outperform standard input PCA bases in jointly approximating both operators and their derivatives. Combining Sobolev-type error bounds, empirical sampling analysis, and elliptic PDE experiments, we demonstrate that DIS achieves higher reconstruction accuracy and generalization with lower rank and fewer training samples. The theoretical bounds are tight and practically actionable.

Conventional high-dimensional analysis methods struggle to simultaneously capture the coupling between structure and dynamics in complex many-body systems. This work proposes the Time Derivative (TiDe) space framework, which constructs higher-order time derivatives directly from temporal observational data as intrinsic dimensions, enabling joint unsupervised learning of structure and dynamics without dimensionality reduction. By naturally embedding physical information into high-dimensional representations, TiDe achieves both interpretability and computational robustness. Experiments on molecular dynamics simulations and experimental trajectory data demonstrate that TiDe efficiently uncovers key dynamical patterns in complex systems, significantly outperforming existing approaches.

Traditional neural networks suffer from limited interpretability and weak theoretical foundations. Method: This paper proposes a novel machine learning paradigm grounded in infinite-dimensional Hilbert spaces, centering on linear operators. It integrates reproducing kernel Hilbert spaces (RKHS), spectral operator learning, wavelet representations, scattering transforms, and Koopman operator theory to formulate learning tasks as sampling, approximation, and dynamical inference in infinite-dimensional function spaces. Contribution/Results: We establish the first unified Hilbert-space-theoretic framework bridging spectral learning and symbolic reasoning. The approach significantly enhances mathematical rigor and model interpretability by grounding learning in well-defined functional-analytic principles. Moreover, it provides a rigorous mathematical foundation and new methodological pathways for deep interdisciplinary integration between signal processing and machine learning—enabling principled analysis of structured data, hierarchical feature extraction, and nonlinear dynamical system modeling.

Scientific problems are often formulated in infinite-dimensional function spaces (e.g., PDE solution operators), whereas mainstream deep learning models are restricted to finite-dimensional mappings, limiting their generalization in scientific computing. To address this, we propose a systematic paradigm for extending classical neural networks (e.g., CNNs, Transformers) into *neural operators*, introducing— for the first time—the four fundamental design principles for function-space mappings, enabling low-intrusion, analytically tractable architectural migration. Our method integrates Fourier/wavelet-based operators, multi-scale attention, and discretization-invariance constraints, augmented by spectral-domain projection and mesh-agnostic parameterization. Evaluated on benchmarks including Navier–Stokes and Darcy flow equations, our models achieve substantial improvements in generalization across varying geometries, boundary conditions, and material coefficients; they also deliver 3.2× inference speedup and reduce generalization error by 47%.

Weak generalization and poor extrapolation of coordinate-based MLPs in periodic signal modeling motivate this work. We propose the first neural function architecture explicitly designed for periodic signals, embedding implicit periodic inductive bias directly into the network structure. Our approach integrates learnable periodic embeddings, a phase-alignment module, and frequency-domain regularization, while jointly training with physics-informed constraints from differential equations and real-world time-series data. Compared to baselines including SIREN and Fourier Feature networks, our method achieves an average 37.2% reduction in extrapolation error across tasks—namely, learning solutions to periodic PDEs, and interpolation and extrapolation of real temporal sequences. It significantly enhances extrapolation robustness and physical consistency. This work establishes a new paradigm for improving the generalization capability of continuous neural representations in modeling periodic dynamical systems.

This work proposes a learnable framework for adaptive orthogonal bases that overcomes the rigidity of traditional fixed bases—such as Fourier or wavelet bases—in capturing data-specific structures. The target basis is treated as a point on the Lie manifold of the orthogonal group and is obtained by continuously evolving a reference basis along a path defined by an ordinary differential equation induced by a finite-rank skew-adjoint integral operator, parameterized via neural networks. Theoretically, it is shown that rank-2 generators suffice to densely approximate any orthogonal basis in the operator topology, ensuring both universality and flexibility. Experiments demonstrate successful adaptation of the Fourier basis into data-driven principal components, eigenfunctions of operators, and dynamic modes of physical systems, confirming the method’s effectiveness and broad applicability.

To address poor physical consistency, inadequate high-frequency process modeling, and weak stability in high-resolution, long-term ocean dynamics forecasting, this paper proposes FNOtD: a Fourier neural operator (FNO) variant that explicitly incorporates dispersion relation constraints and jointly learns spatiotemporal integration kernels, enabling multi-scale wave propagation modeling guided by temporal Fourier modes. This design enhances the model’s capacity to represent high-frequency dynamical processes, improves long-term prediction stability, and strengthens adherence to underlying physical laws. Experiments demonstrate that FNOtD achieves accuracy comparable to state-of-the-art numerical models on high-resolution ocean forecasting tasks, reduces computational cost by an order of magnitude, and lowers prediction error by over 40% beyond 50 time steps compared to the standard FNO.

Modeling, classification, and forecasting of large-scale spatiotemporal data from high-dimensional nonlinear complex systems—such as brain activity, climate, and ecosystems—remain challenging due to the limited representational capacity of conventional dimensionality reduction and phase-space reconstruction methods. Method: We propose a geometric vector field analysis framework on discrete measure spaces, introducing for the first time a two-parameter family of vector field metrics applicable to spatiotemporal functions defined on graphs and simplicial complexes. This framework unifies representations of scalar fields, gradient fields, and multivalued fields, transcending classical attractor-geometric limitations. By integrating vector field representation theory, discrete differential geometry, and multidimensional scaling (MDS), it enables model-free, efficient dimensionality reduction, modal decomposition, phase-space reconstruction, and attractor characterization. Results: Extensive validation on biological and physical simulation datasets demonstrates substantial improvements in dynamical system analysis capability, particularly in capturing nonlinear, multiscale spatiotemporal structures.

This work proposes a novel Toeplitz filtering framework for accurately estimating the spectral properties of linear evolution operators—such as Koopman or transfer operators—from equation-free equilibrium trajectory data. By introducing Toeplitz structure into spectral estimation and incorporating structural priors like self-adjointness or skew-symmetry on the infinitesimal generator, the method enables efficient recovery of eigenvalues, eigenfunctions, and spectral measures. Coupled with a primal-dual statistical learning algorithm, the framework achieves both statistical consistency and computational efficiency. Numerical experiments demonstrate that the approach precisely reconstructs fine-grained spectral structures in both deterministic and chaotic dynamical systems—features often missed by conventional data-driven techniques.

为解决时间序列预测中Koopman空间数学不一致性和低秩结构捕捉不足的问题，提出K^2SVD方法，通过优化Hilbert-Schmidt目标学习Koopman算子的主要奇异函数，并结合卡尔曼滤波进行推断。