closed-form parameter derivation

Many equations in science lack closed-form solutions, and existing symbolic regression methods rely on input-output data, making it difficult to solve equations using only their mathematical form. This work proposes a Symbolic Equation Solver (SES), which, for the first time, frames equation solving as a data-free optimization problem over a differentiable symbolic model. SES constructs an objective function solely from the equation’s structure and its initial or boundary conditions, without requiring any observed data. By leveraging symbolic expression generation techniques, the method directly recovers compact and accurate analytical solutions from the equation itself. It demonstrates success across diverse equation types—including algebraic equations, those involving transcendental terms, ordinary differential equations, and partial differential equations—thereby overcoming the traditional dependence of symbolic regression on training data.

This study investigates the solution structure and multiplicity of maximum likelihood estimators in moving average (MA) and autoregressive (AR) time series models, with a focus on classifying critical points that yield non-invertible models. For the first time, algebraic statistics and numerical algebraic geometry are systematically employed to analyze the likelihood equations, leading to closed-form algebraic solutions for parameters in certain cases. The paper also introduces composite likelihood as an alternative estimation strategy. Combining theoretical analysis with simulation experiments, the authors demonstrate that the proposed algebraic methods achieve superior accuracy and numerical stability compared to conventional optimization algorithms, thereby offering a novel theoretical framework and computational toolkit for parameter estimation in time series analysis.

This study addresses the high computational cost and low efficiency of traditional methods for parameter estimation in differential equation models. To overcome these limitations, the authors propose a penalized likelihood framework based on the generalized profiling (parameter cascading) approach, which directly embeds ordinary differential equations (ODEs) into the objective function, thereby avoiding repeated numerical integration. This strategy preserves dynamic consistency while substantially improving estimation efficiency and numerical stability. As a key contribution, the work provides an open-source, reproducible Jupyter Notebook tutorial with complete code implementations covering multiple ODE modeling examples, effectively lowering the barrier to adopting advanced parameter estimation techniques and facilitating their broader application in both research and education.

Existing symbolic regression methods rely heavily on heuristic model selection, regularization, and search strategies, lacking theoretical foundations and performance guarantees. Method: This paper introduces a novel Bayesian inference–based paradigm for symbolic regression, replacing heuristics with a probabilistic framework where model discovery is formulated as posterior distribution inference. It integrates information-theoretic principles (e.g., Minimum Description Length) and statistical physics concepts (e.g., variational approximation) to naturally balance model complexity and goodness-of-fit. Crucially, it emphasizes model ensembling over selecting a single optimal expression, enabling principled uncertainty quantification and theoretically grounded generalization bounds. Results: Extensive experiments demonstrate that our approach significantly outperforms state-of-the-art symbolic regression methods in equation discovery accuracy, physical consistency, and robustness across diverse datasets.

This work addresses the abrupt changes in the number of real critical points of likelihood equations in algebraic statistical models, which are governed by the model’s non-properness locus—a set characterizing data configurations for which the system admits solutions at infinity. The paper introduces, for the first time, an efficient method to compute this non-properness locus by integrating tools from algebraic geometry, real root classification, and discriminant variety theory. The authors rigorously establish the correctness of their approach and develop a corresponding detection algorithm. Experimental results demonstrate that the proposed method significantly outperforms existing techniques in computational efficiency, offering a practical and scalable solution for data classification tasks based on the number of real solutions.

This work addresses a key limitation in existing gradient-based symbolic regression methods, which struggle to incorporate operations such as division and logarithms due to singularities or domain restrictions on the real line, thereby constraining the search space. To overcome this, the paper introduces complex-domain optimization into symbolic regression for the first time. By performing gradient descent in the complex plane, the optimization trajectory can circumvent singularities on the real axis, enabling stable and unconstrained use of a broad class of nonlinear operations. This approach effectively avoids degeneracies inherent in real-valued optimization, substantially expanding the set of learnable symbolic expressions. The method successfully reconstructs target functions with singularities on standard benchmarks and accurately recovers singular behaviors from experimental frequency response data.

This work proposes ViSA-R2, a novel framework designed to recover analytical solutions from 2D steady-state physical field visualizations and their derivatives, thereby enabling AI-driven scientific reasoning. The method establishes an end-to-end pipeline that emulates the physicist’s reasoning process—structured as structure identification, analytical hypothesis formulation, parameter derivation, and consistency verification—and introduces a self-validating, solution-centered chain-of-thought mechanism. Built upon Qwen3-VL (8B) and SymPy, the study also releases ViSA-Bench, the first benchmark supporting vision-language models for symbolic reasoning in physics, evaluated through multi-dimensional metrics including numerical accuracy, structural similarity, and character-level precision. Experiments demonstrate that ViSA-R2 significantly outperforms both open- and closed-source vision-language models across 30 linear steady-state scenarios, achieving high-fidelity symbolic expression inference.

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 proposes the AI-Kolmogorov framework, which introduces symbolic regression systematically into probability density estimation to address the Symbolic Density Estimation (SymDE) problem. The method decomposes complex distributions through clustering or probabilistic graphical models in a multi-stage process, sequentially integrating support set estimation, nonparametric density estimation, and symbolic regression to construct interpretable analytic expressions of probability densities. Evaluated on synthetic mixture models, multivariate normal distributions, and exotic distributions from high-energy physics, the framework successfully recovers or uncovers their underlying mathematical structures, enabling both interpretable modeling and structural discovery for complex probability distributions.

This work proposes a data-driven approach based on symbolic regression to automatically discover concise and highly accurate parametric expressions for the implied volatility surface. The method directly searches market data for analytical forms of total implied variance as a function of log-moneyness and time to maturity, without assuming any predefined functional structure. As the first application of symbolic regression to implied volatility modeling, the resulting formulas are both interpretable and compact, achieving fitting accuracy and model parsimony that rival or even surpass those of the widely used Stochastic Volatility Inspired (SVI) framework.