monotone-spline ccif parameterization

Existing approaches to modeling cumulative conditional intensity functions using monotonic neural networks are hindered by convexity constraints, saturation bottlenecks, and structural limitations, impeding their ability to capture complex temporal dynamics effectively. This work proposes the Monotonic Alternating Splines (MAS) framework, which decouples interpolation and extrapolation components to enable flexible monotonic function approximation. By doing so, MAS maintains high fitting accuracy while significantly enhancing generalization capabilities. The method overcomes the structural constraints inherent in conventional monotonic neural networks and theoretically eliminates irreducible approximation error. Empirical evaluations demonstrate that MAS consistently outperforms state-of-the-art methods on both synthetic and real-world datasets, achieving superior performance with remarkable efficiency and accuracy.

Existing PDE foundation models rely heavily on large-scale Transformer architectures, incurring prohibitive parameter counts and computational costs. To address this, we propose SPUS—the first unified neural operator foundation model built upon a lightweight residual U-Net architecture. Our key contributions are threefold: (i) the first integration of residual U-Net into PDE foundation modeling; (ii) a physics-informed autoregressive pretraining strategy that explicitly emulates numerical PDE solvers to learn conservation laws and dynamical evolution; and (iii) joint pretraining on multi-physics fluid PDEs coupled with few-shot transfer mechanisms. Experiments demonstrate that SPUS achieves state-of-the-art generalization across six unseen PDE tasks, reducing parameter count by one to two orders of magnitude compared to mainstream approaches. Moreover, it adapts rapidly to novel equations with only a few fine-tuning samples.

This work addresses the longstanding challenge that existing spline interpolation methods struggle to simultaneously achieve exact interpolation and strict arc-length parameterization. The paper proposes an iterative optimization algorithm capable of constructing spline curves in arbitrary-dimensional spaces that exactly interpolate prescribed data points while being strictly parameterized by arc length. This approach represents the first method to unify exact interpolation with rigorous arc-length parameterization, thereby overcoming the traditional trade-off between parametrization quality and interpolation accuracy inherent in conventional splines. Numerical experiments in two-dimensional space demonstrate the algorithm’s effectiveness and high precision, highlighting its promising applications in geometric modeling, trajectory generation, and related fields.

To address the poor interpretability, weak extrapolation capability, and low computational efficiency of black-box models in scientific computing and general-purpose machine learning, this paper proposes a class of physics-informed parametric matrix models. These models incorporate structural priors derived from matrix equations expressed algebraically, differentially, or integrally—marking the first integration of physical modeling paradigms into general function approximation. We theoretically establish their universal approximation property. Crucially, differential and integral operators are explicitly embedded in the model architecture, enabling principled extrapolation beyond training input domains. Parameters are optimized via empirical data-driven learning, yielding models that simultaneously achieve high predictive accuracy, computational efficiency, and full analytical tractability. Empirical evaluation across diverse scientific computing tasks—including partial differential equation solving—and general machine learning benchmarks demonstrates substantial improvements in extrapolation robustness and generalization performance.

This work addresses the lack of efficient parametric solution methods for zero-dimensional parametric polynomial systems that exhibit favorable specialization properties. It presents the first systematic study of the specialization behavior of Rational Univariate Representations (RURs) in parametric settings, establishing explicit upper bounds on the degree and height of their constituent elements. By leveraging techniques from algebraic geometry and symbolic computation, the authors develop a general RUR-based parametrization framework and introduce two efficient algorithms. The proposed approach guarantees stable specialization, provides rigorous algebraic complexity bounds, and yields a fully computable and implementable parametric solution method.

This work addresses the lack of unified CPU/GPU backend support and symbolic-numeric integration in existing tools for multivariate polynomial modeling. We propose a MATLAB-based toolbox offering three interfaces—MPOLY, MPOLY_GPU, and MPOLY_HP—that enable construction, algebraic manipulation, differentiation, LaTeX export, and interoperability with YALMIP and SOSTOOLS for polynomials and polynomial matrices. Key innovations include a variable metadata mechanism ensuring safe handling of mixed-variable expressions, a dedicated layer for affine normal vector computation, and a unified CPU/GPU architecture that seamlessly blends symbolic and numeric processing. Experimental results demonstrate that MPOLY is well-suited for lightweight interactive tasks, MPOLY_HP achieves higher efficiency in medium- to large-scale affine computations, and the proposed randomized log-determinant algorithm exhibits significant advantages in sparse, large-scale settings.

This work addresses the problem of inferring models in systems biology that must satisfy monotonicity constraints—requiring certain inputs to exert a consistent positive or negative influence on outputs—by introducing, for the first time, a combination of uninterpreted functions from SMT solving with explicit monotonicity constraints. The authors propose an efficient lazy lemma generation mechanism, integrated with quantifier-aware encoding and an active quantifier instantiation strategy. Evaluated on multiple systems biology benchmarks, the approach significantly outperforms domain-specific tools such as Bonesis (based on Answer Set Programming) and AEON (based on Binary Decision Diagrams), demonstrating superior scalability and solving efficiency, particularly on high-arity and structurally complex instances.

研究解决了深度样条叠加网络中逼近精度与深度稳定性之间的矛盾，通过精确的层平衡和预算兼容的离散化方法来实现。

This work addresses the challenge of parameter estimation in physical system simulation models arising from complex residual distributions. The authors propose an end-to-end estimation framework that employs embedded normalizing flows to map intricate residuals onto a simple base distribution. Indirect constraints are imposed on this base distribution through empirical likelihood under moment conditions. Model and flow parameters are jointly optimized via implicit differentiation combined with first-order gradient methods. By innovatively integrating normalizing flows with constrained empirical likelihood, the approach establishes an information-theoretically interpretable and computationally tractable framework. The resulting inverse transformation serves as an invertible surrogate model, enhancing both the accuracy and efficiency of parameter estimation while enabling quantification of model bias and sensitivity analysis.

This work addresses the limitations of traditional Kolmogorov–Arnold Networks (KANs), which rely on fixed-grid B-spline bases and struggle to approximate nonsmooth or multiscale functions effectively. The authors introduce, for the first time, fractal geometry into neural function approximation by proposing Pure FI-KAN and Hybrid FI-KAN architectures. These models employ learnable fractal interpolation function (FIF) bases constructed via iterated function systems (IFS) and incorporate a differentiable fractal dimension as an adaptive parameter to dynamically align the Hölder regularity of the basis functions with that of the target function. Empirical results demonstrate substantial improvements over conventional KANs: on Hölder continuity benchmarks, performance gains range from 1.3× to 33×; mean squared error on fractal targets is reduced by 6.3×; and accuracy in approximating solutions to nonsmooth partial differential equations improves by up to 79×.