integral equation solving

The discovery of first integrals in dynamical systems has long been hindered by the scarcity of high-quality data and reliance on mathematical intuition. This work proposes FISolver, which introduces a novel inverse generation algorithm to construct a large-scale dataset of differential equation–first integral pairs. By integrating supervised fine-tuning with reinforcement learning guided by Levenshtein distance, FISolver optimizes a small mathematical language model. Through reward shaping and synthetic data strategies, it achieves efficient generalization from sparse examples to complex families of problems. Experiments demonstrate that FISolver significantly outperforms both large mathematical language models and commercial solvers such as Mathematica on challenging benchmarks, while substantially reducing computational costs.

This work addresses the challenge that existing proximal causal inference methods cannot identify the interventional joint distribution involving all proxy variables. To overcome this limitation, we propose, for the first time, an extended bridge function and establish a theoretical identifiability result for this distribution, integrating it into a kernel-based proximal causal inference framework. By introducing a synergistic mechanism between the interventional kernel and the extended bridge function, we derive novel identification conditions for the interventional joint distribution and develop a general, computable kernelized proximal identification algorithm. This contribution not only broadens the theoretical foundations of proximal causal inference but also provides a practical tool for estimating causal effects in settings with high-dimensional proxy variables.

Modeling strongly path-dependent financial derivatives—such as exotic options and variance instruments—remains challenging due to the non-Markovian nature of path-dependent functionals. Method: This paper introduces the “occupied process” framework, augmenting the original process $X$ with its occupation measure flow $O$ to form a Markovian lifted system $(O,X)$. It defines the novel “occupation derivative”, unifying functional Itô calculus and mean-field derivatives, and recasts a broad class of path-dependent PDEs as parabolic equations in the occupation measure time variable. Contribution/Results: The framework enables an Itô calculus tailored to path occupation-time functionals and extends the Feynman–Kac formula accordingly. It yields closed-form solutions to local-time-driven optimal stopping problems, with direct applications to corridor variance swap pricing and path-dependent volatility modeling. By bridging stochastic analysis, mean-field theory, and financial mathematics, this work substantially expands both the theoretical foundations and practical applicability of path-dependent stochastic modeling.

This work addresses the lack of interpretability and resolution-invariant modeling capability of diffusion models in infinite-dimensional function spaces—such as images, time series, and probability density functions (PDFs). To this end, we extend stochastic optimal control theory to infinite dimensions. Our method establishes, for the first time, a rigorous equivalence between infinite-dimensional Doob h-transforms and stochastic optimal control; overcomes the fundamental challenge of undefined densities in infinite dimensions by constructing diffusion bridges without explicit density assumptions; and jointly leverages variational inference and functional optimization to directly compute optimal transport paths in function space. Experiments demonstrate that the proposed framework achieves resolution invariance in both distributional bridge learning and sampling tasks. It significantly improves fidelity and interpretability across diverse applications—including image generation, time-series interpolation, and PDF modeling—without dependence on spatial or temporal discretization.

Modeling individualized local dynamics in high-noise, multivariate longitudinal clinical data using ordinary differential equations (ODEs) remains challenging due to poor robustness and identifiability. Method: We propose a local ODE modeling paradigm that initializes ODE solutions at each observed time point, generating and aggregating multiple local ODE trajectories per observation. The framework jointly learns a low-dimensional dynamic latent space and patient-specific ODE parameters via a variational autoencoder architecture, differentiable programming, and baseline-feature-driven ODE parameterization—enabling end-to-end interpretable modeling. Contribution/Results: Evaluated on real-world spinal muscular atrophy cohort data and synthetic experiments, our method significantly improves short-term local prediction accuracy of health-state changes over global regression models (p < 0.01) and enhances dynamic interpretability. It establishes a novel paradigm for personalized disease progression modeling grounded in mechanistic, observation-driven ODE learning.

Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are well studied: here we show how similar workflows can be used to recover functions from data. Specifically, we embed neural networks into the PDE and show how, as they are trained on data, they can approximate unknown functions with arbitrary accuracy. Using nonlocal aggregation-diffusion equations as a case study, we recover interaction kernels and external potentials from steady state data. Specifically, we investigate how a wide range of factors, such as the number of available solutions, their properties, sampling density, and measurement noise, affect our ability to successfully recover functions. Our approach is advantageous because it can utilise standard parameter-fitting workflows, and in that the trained PDE can be treated as a normal PDE for purposes such as generating system predictions.

This study addresses the need for a unified understanding of the relationships and generalization capabilities across data-driven modeling paradigms, ranging from classical inverse problems to modern neural operators. By integrating inverse problem theory, sparse identification of dynamical systems, neural ordinary differential equations, and neural operators—and further incorporating the philosophical notion of “mechanism” from philosophy of science—the authors construct a cohesive analytical framework. The work demonstrates that genuine mechanistic discovery and robust cross-scenario generalization are achievable only when models recover concise differential equation structures underlying the observed data. This perspective clarifies the fundamental connections among diverse modeling approaches and underscores the critical role of mechanistic interpretability in enabling reliable generalization, thereby offering a theoretical foundation for the categorization, selection, and design of scientific machine learning models.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.

Identifying nonlocal kernel functions in integro-differential equations from sparse and noisy spatiotemporal observations constitutes an ill-posed inverse problem. This work proposes a physics-informed framework based on differentiable solvers that leverages a novel Kolmogorov–Arnold Network (KAN) to directly learn the kernel from data and subsequently recover its analytic expression via symbolic regression. The approach innovatively incorporates hard constraints through Bernstein-polynomial-based MC-KAN and soft constraints via Chebyshev-basis-based Cheb-KAN—both introduced here for the first time in kernel discovery—and demonstrates that MC-KAN exhibits superior robustness under high-dimensional, sparse, and noisy conditions. Validated on a 1D Volterra equation, a viscoelastic wave equation, and a 2D anisotropic nonlocal reaction–diffusion equation, MC-KAN significantly outperforms Cheb-KAN in accurately recovering the true kernel structure and reconstructing the solution field.

This work addresses the limitations of traditional methods for discovering governing equations from observational data, which often rely on manually prescribed basis functions and fail due to missing terms or complex dynamics, as well as pure symbolic regression approaches that are sensitive to noise and yield redundant expressions. The authors propose AutoSINDy, a novel three-stage automated framework that synergistically integrates symbolic regression with sparse identification. It leverages PySR to generate candidate functions from resampled data, applies collinearity-aware decomposition and selection to refine expressions, and employs SINDy for sparse regression—eliminating the need for predefined basis functions. Evaluated across diverse nonlinear systems, AutoSINDy recovers the true equations with 92.8% accuracy, significantly outperforming baseline methods while achieving superior predictive accuracy, enhanced generalization under high noise, and reduced symbolic complexity.