operator splitting

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This paper addresses the computation of the greatest common right divisor (GCRD) for linear ordinary differential operators with non-rational coefficients—such as parametric transcendental functions—where classical algorithms assume rational-function coefficients. The proposed symbolic algorithm is grounded in the determinant definition of the differential resultant; it explicitly constructs the resultant matrix and leverages Maple’s DEtools package and symbolic computation capabilities for efficient GCRD computation, naturally accommodating parametric coefficients. Key contributions include: (i) the first implementation of a GCRD algorithm for non-rational coefficient operators within a general-purpose symbolic computation platform; (ii) a practical, parameter-aware tool for analyzing parametric differential systems; and (iii) successful application to the construction of commuting differential operators, thereby empirically validating the algorithm’s correctness and effectiveness.

To address spectral information loss and multiplicative structure degradation in finite-dimensional approximations of the Koopman operator—caused by heuristic selection of observable function dictionaries—this paper proposes a novel structure-guided Dynamic Mode Decomposition (DMD) framework. The core innovation lies in the first explicit incorporation of the Koopman operator’s intrinsic multiplicative structure into the DMD modeling process, achieved via constrained matrix optimization that jointly enables adaptive observable selection and structural preservation. The method enjoys theoretically guaranteed convergence and significantly improves spectral estimation accuracy and robustness to noise. Experiments on a simple pendulum, the Lorenz system, and real-world fluid flow data demonstrate superior spectral fidelity and stability compared to standard DMD.

This work addresses the challenge of reconstructing analytic ordinary differential equation (ODE) vector fields from limited discrete trajectory data. Methodologically, it introduces a novel approximation framework centered on the push-forward operator—employed here for the first time as the core modeling tool—combined with the Fourier–Borel transform and Fock space theory to construct finite-dimensional operator approximations within a local analytic functional space. Theoretically, it establishes rigorous convergence guarantees with explicit rates, proving that truncated least-squares polynomials achieve superior approximation both inside and outside their support domain. Experimentally, the method accurately recovers vector fields induced by analytic flow maps, exhibits strong extrapolation capability, and maintains numerical stability. Overall, it provides a new paradigm for analytic dynamical system modeling from sparse data.

Quantum linear solvers (e.g., HHL) applied to discretized PDEs suffer from complexity scaling polynomially with the condition number κ, which typically grows polynomially with system size N—constituting a fundamental bottleneck. Method: We propose the first κ-independent quantum PDE solver framework, introducing wavelet bases as auxiliary coordinates and integrating diagonal preconditioning to render the preconditioned system’s condition number independent of N. Building upon this, we design a quantum algorithm with polylogarithmic complexity. Contribution/Results: We rigorously prove the overall query and gate complexity is poly(log N). Numerical experiments confirm substantial reduction in effective condition number across diverse PDEs—including elliptic, parabolic, and convection-diffusion equations—and demonstrate efficient extraction of solution features from the output quantum state. This work establishes the first theoretically sound and practically viable κ-independent quantum framework for PDE solving.

This work addresses the unclear relationship between continuous theory and discrete implementation in neural operators for solving partial differential equations, particularly concerning stability and discretization error. For the first time, rigorous discretization error bounds are established for State-Space Neural Operators (SS-NOs) and Fourier Neural Operators (FNOs). By leveraging functional analysis, the regularity of solutions is explicitly linked to input discretization, and Input-to-State Stability (ISS) theory is introduced to quantify how discretization affects stability in the continuous domain. Numerical experiments on one- and two-dimensional benchmark problems validate the tightness of the derived theoretical bounds, demonstrating that SS-NOs exhibit both robustness and numerical stability across varying resolutions.

This work addresses the challenge of high errors in neural surrogate models for stiff differential-algebraic equations (DAEs), which arise due to algebraic residuals being amplified by stiffness or reliance on costly numerical integration. The authors propose an extended Newton implicit layer that jointly enforces algebraic consistency and quasi-steady-state dimensionality reduction within a single differentiable solve. By predicting only the slow-varying states, the method accurately recovers both fast dynamics and algebraic variables while reducing output dimensionality to the slow subspace. This is the first approach to integrate physics-guided dimensionality reduction with implicit differential operator learning without simulation-based training. Leveraging the implicit function theorem, it derives gradients accounting for stiff coupling, supports compositional modeling of multi-component cascaded systems, and offers theoretical convergence guarantees. Evaluated on a 21-state power inverter DAE, it achieves a mere 1.42% error—significantly outperforming penalty methods (39.3%)—and composes two models into a 44-state system without retraining, yielding 0.72–1.16% error, zero algebraic residual, and 90% in-distribution coverage via conformal prediction.

This work systematically compares neural operators and polynomial surrogates within a unified framework to reduce the computational cost of repeatedly solving parametric partial differential equations, with a focus on how surrogate performance depends on the regularity of input fields. Employing methods including Fourier Neural Operators, reduced-basis neural operators (trained with respect to $L^2_\mu$ and $H^1_\mu$ losses), reduced-basis sparse grids, and tensor-train polynomials, the study reveals for the first time that surrogate efficacy is highly sensitive to input smoothness: polynomial approaches exhibit markedly superior data efficiency when inputs are smooth ($s \geq 2$), whereas Fourier Neural Operators converge fastest for rough inputs ($s \leq 1$). Furthermore, incorporating derivative information during training significantly enhances accuracy and efficiency in low-data regimes.

This work addresses the efficient computation of partial Weyl closures for parametric differential operator systems—a crucial preprocessing step in symbolic integration and related algorithms—by transforming rational coefficients into polynomial ones. We propose a novel algorithm that generalizes the Rabinowitsch trick within a noncommutative algebraic framework to effectively approximate the Weyl closure of holonomic modules with respect to a subset of variables. The resulting module is guaranteed to be contained within the true closure. To our knowledge, this is the first method to achieve an efficient approximation of partial Weyl closures, substantially avoiding the high computational complexity of exact approaches. Implemented in Julia within the MultivariateCreativeTelescoping.jl package, our method demonstrates significant speedups over exact implementations in Singular and Macaulay2 across multiple test cases, while preserving correctness through guaranteed containment.

This work addresses the high memory and computational complexity typically associated with solving three-dimensional partial differential equations on Cartesian grids. By exploiting tensor-product structure, the proposed method decomposes the 3D operator into one-dimensional banded kernels aligned with coordinate axes, thereby avoiding explicit assembly of the global matrix and enabling a matrix-free solution strategy. Within a unified framework that integrates diverse numerical approaches—including Kronecker product algebra, compact finite differences, isogeometric analysis, and direct diagonalization—the study systematically identifies three key techniques: multi-right-hand-side reshaping, sum factorization, and pencil-style MPI decomposition. These innovations collectively enhance hardware affinity and parallel scalability, reducing algorithmic complexity to O(N) and storage requirements to O(Nₓ + Nᵧ + N_z), thus enabling efficient large-scale 3D PDE simulations.