Abstract We study the problem of exact equation discovery, i.e., identifying symbolic equations that perfectly describe relationships in noise-free data. While most existing approaches focus on approximate recovery from noisy measurements, we consider settings in which exact correctness is required. This setting is closely related to methods that infer symbolic relations, such as recurrence equations or generating functions, from finite data. We show that exact equation discovery can be formulated as the computation of the vanishing ideal of the observed data and leverage Gröbner bases as an effective algorithmic tool. Building on this connection, we introduce MoadeeB, a new algorithm for discovering exact equations over integers and rational numbers. We evaluate MoadeeB in a large-scale empirical study on more than 30,000 integer sequences from the Online Encyclopedia of Integer Sequences (OEIS), focusing on the reconstruction of known recurrences and the discovery of previously undocumented ones. We compare against state-of-the-art symbolic regression and program synthesis approaches, as well as approaches from experimental mathematics and computer algebra that infer symbolic relations directly from finite sequence prefixes. The results show that MoadeeB achieves competitive or superior performance across these method classes, while additionally enabling the discovery of exact equations beyond the scope of existing approaches.

This paper introduces an innovative deep learning-based optimization method specifically designed for data derived from stochastic processes . Addressing the prevalent issue of rapid overfitting in real-world scenarios with limited historical data , our approach focuses on denoising optimization. The method effectively balances the simultaneous optimization of latent data representation and target variables, leading to enhanced model performance. We rigorously test our approach using five diverse real-world datasets. Our study is structured into three parts: an ablation study to validate the individual components of our method, a statistical analysis using the Wilcoxon rank-sum test to confirm the superiority of our method against five research hypotheses, and a detailed exploration of parameter visualization and fine-tuning. The comprehensive evaluation demonstrates that our method not only outperforms existing techniques but also significantly contributes to the advancement of deep learning models for stochastic processes. The findings underscore the potential of our method as a robust solution to the challenges in modeling stochastic processes with deep learning , offering new avenues for efficient and accurate predictions .

Abstract Ordinary differential equations (ODEs) are a widely used formalism for the mathematical modeling of dynamical systems, a task omnipresent in scientific domains. The paper introduces a novel method for inferring ODEs from data, which extends ProGED, a method for equation discovery that allows users to formalize domain-specific knowledge as probabilistic context-free grammars and use it for constraining the space of candidate equations. The extended method can discover ODEs from partial observations of dynamical systems, where only a subset of state variables can be observed. To evaluate the performance of the newly proposed method, we perform a systematic empirical comparison with alternative state-of-the-art methods for equation discovery and system identification from complete and partial observations. The comparison uses Dynobench, a set of ten dynamical systems that extends the standard Strogatz benchmark. We compare the ability of the considered methods to reconstruct the known ODEs from synthetic data simulated at different temporal resolutions. We also consider data with different levels of noise, i.e., signal-to-noise ratios. The improved ProGED compares favourably to state-of-the-art methods for inferring ODEs from data regarding reconstruction abilities and robustness to data coarseness, noise, and completeness.

We introduce MLFMF, a collection of data sets for benchmarking recommendation systems used to support formalization of mathematics with proof assistants. These systems help humans identify which previous entries (theorems, constructions, datatypes, and postulates) are relevant in proving a new theorem or carrying out a new construction. Each data set is derived from a library of formalized mathematics written in proof assistants Agda or Lean. The collection includes the largest Lean 4 library Mathlib, and some of the largest Agda libraries: the standard library, the library of univalent mathematics Agda-unimath, and the TypeTopology library. Each data set represents the corresponding library in two ways: as a heterogeneous network, and as a list of s-expressions representing the syntax trees of all the entries in the library. The network contains the (modular) structure of the library and the references between entries, while the s-expressions give complete and easily parsed information about every entry.We report baseline results using standard graph and word embeddings, tree ensembles, and instance-based learning algorithms. The MLFMF data sets provide solid benchmarking support for further investigation of the numerous machine learning approaches to formalized mathematics. The methodology used to extract the networks and the s-expressions readily applies to other libraries, and is applicable to other proof assistants. With more than $250\,000$ entries in total, this is currently the largest collection of formalized mathematical knowledge in machine learnable format.

Equation discovery, also known as symbolic regression, is a machine learning task of inducing closed-form equations from data and background knowledge. The latter takes various forms. Domain-specific knowledge can constrain the space of candidate equations to those that make sense in the scientific or engineering domain of use. Cross-domain knowledge, on the other hand, imposes general rules for model acceptability, such as parsimony, understandability, or consistency of the equations with the dimensional units of the variables. In this paper, we propose using attribute grammars to ensure the induced equations' dimensional consistency. Attribute grammars are flexible enough to combine cross-domain knowledge on dimensional consistency with domain-specific knowledge expressed as a probabilistic context-free grammar. At the same time, we show that attribute grammars can be efficiently transformed into probabilistic context-free grammars for equation discovery with existing algorithms. Finally, we provide empirical evidence that attribute grammars ensuring dimensional consistency of equations can significantly improve the performance of equation discovery on the standard set of a hundred Feynman benchmarks.

Abstract We propose an approach to symbolic regression based on a novel variational autoencoder for generating hierarchical structures, HVAE. It combines simple atomic units with shared weights to recursively encode and decode the individual nodes in the hierarchy. Encoding is performed bottom-up and decoding top-down. We empirically show that HVAE can be trained efficiently with small corpora of mathematical expressions and can accurately encode expressions into a smooth low-dimensional latent space. The latter can be efficiently explored with various optimization methods to address the task of symbolic regression. Indeed, random search through the latent space of HVAE performs better than random search through expressions generated by manually crafted probabilistic grammars for mathematical expressions. Finally, EDHiE system for symbolic regression, which applies an evolutionary algorithm to the latent space of HVAE, reconstructs equations from a standard symbolic regression benchmark better than a state-of-the-art system based on a similar combination of deep learning and evolutionary algorithms.

Multi-label classification (MLC) has recently attracted increasing interest in the machine learning community. Several studies provide surveys of methods and datasets for MLC, and a few provide empirical comparisons of MLC methods. However, they are limited in the number of methods and datasets considered. This paper provides a comprehensive empirical investigation of a wide range of MLC methods on a wealth of datasets from different domains. More specifically, our study evaluates 26 methods on 42 benchmark datasets using 20 evaluation measures. The evaluation methodology used meets the highest literature standards for designing and conducting large-scale, time-limited experimental studies. First, the methods were selected based on their use in the community to ensure a balanced representation of methods across the MLC taxonomy of methods within the study. Second, the datasets cover a wide range of complexity and application domains. The selected evaluation measures assess the predictive performance and efficiency of the methods. The results of the analysis identify RFPCT, RFDTBR, ECCJ48, EBRJ48, and AdaBoost.MH as the best-performing methods across the spectrum of performance measures. Whenever a new method is introduced, it should be compared with different subsets of MLC methods selected according to relevant (and possibly different) evaluation criteria.

Meta learning generalizes the empirical experience with different learning tasks and holds promise for providing important empirical insight into the behavior of machine learning algorithms.In this paper, we present a comprehensive meta-learning study of data sets and methods for multilabel classification (MLC).MLC is a practically relevant machine learning task where each example is labeled with multiple labels simultaneously.Here, we analyze 40 MLC data sets by using 50 meta features describing different properties of the data.The main findings of this study are as follows.First, the most prominent meta features that describe the space of MLC data sets are the ones assessing different aspects of the label space.Second, the meta models show that the most important meta features describe the label space, and, the meta features describing the relationships among the labels tend to occur a bit more often than the meta features describing the distributions between and within the individual labels.Third, the optimization of the hyperparameters can improve the predictive performance, however, quite often the extent of the

Extractive summarization of text documents deals with automatic creation of a summary by combining the most salient sentences extracted from the original text document into a more concise form. In this paper, we introduce a novel graph-based method for extractive summarization that transforms a given text into a graph of interconnected sentences and employs computationally efficient selectivity measure to measure the importance of graph nodes. In turn, the text summary is build upon the sentences corresponding to the most important graph nodes. The edges in the graph are based on three measures of similarity between sentences, i.e., Mihalcea’s, Jaccard and Cosine. The first among the three leads to best performance, which also compares favorably to the performance of three alternative summarization methods of TF-IDF, centroid and TextRank. The results confirm that selectivity of nodes in a text graph build using the Mihalcea’s similarity between sentences is a computationally efficient and well performing method for extractive summarization.

Equation discovery, also known as symbolic regression, is a type of automated modeling that discovers scientific laws, expressed in the form of equations, from observed data and expert knowledge. Deterministic grammars, such as context-free grammars, have been used to limit the search spaces in equation discovery by providing hard constraints that specify which equations to consider and which not. In this paper, we propose the use of probabilistic context-free grammars in equation discovery. Such grammars encode soft constraints, specifying a prior probability distribution on the space of possible equations. We show that probabilistic grammars can be used to elegantly and flexibly formulate the parsimony principle, that favors simpler equations, through probabilities attached to the rules in the grammars. We demonstrate that the use of probabilistic, rather than deterministic grammars, in the context of a Monte-Carlo algorithm for grammar-based equation discovery, leads to more efficient equation discovery. Finally, by specifying prior probability distributions over equation spaces, the foundations are laid for Bayesian approaches to equation discovery.

Automated modeling aims at the induction of mathematical models, both their structure and parameter values, from time-series measurements of observed system variables. In this paper, we address the task of model structure selection, i.e., selecting an optimal structure from a user-specified finite set of alternative model structures, using various approaches to combinatorial search. We propose a mapping of the set of candidate model structures to a fixed-length, vector representation allowing the use of an arbitrary search algorithm as a solver of the structure selection task. We perform a comparative analysis of the performance of thirteen variants of several search algorithms, ranging from ones with high intensification, i.e., focus on neighborhood of the best candidate solutions, to ones with high diversification, i.e., focus on covering the entire search space. The empirical analysis involves eight tasks of reconstructing known models of dynamical systems from synthetic and measured data. The results of the analysis show that search algorithms involving moderate diversification methods have superior performance on the structure selection task. The empirical analysis also reveals that this finding is related to specific properties of the search space of candidate model structures.

Scientists use two forms of knowledge in the construction ofexplanatory models: generalized entities and processes that relatethem; and constraints that specify acceptable combinations of thesecomponents. Previous research on inductive process modeling, whichconstructs models from knowledge and time-series data, has relied onhandcrafted constraints. In this paper, we report an approach todiscovering such constraints from a set of models that have beenranked according to their error on observations. Our approach adaptsinductive techniques for supervised learning to identify processcombinations that characterize accurate models. We evaluate themethod's ability to reconstruct known constraints and to generalizewell to other modeling tasks in the same domain. Experiments with synthetic data indicate that the approach can successfully reconstructknown modeling constraints. Another study using natural data suggests that transferring constraints acquired from one modeling scenario to another within the same domain considerably reduces the amount of search for candidate model structures while retaining the most accurate ones.

People constantly apply acquired knowledge to new learning tasks, but machines almost never do. Research on transfer learning attempts to address this dissimilarity. Working within this area, we report on a procedure that learns and transfers constraints in the context of inductive process modeling, which we review. After discussing the role of constraints in model induction, we describe the learning method, MISC, and introduce our metrics for assessing the cost and benefit of transferred knowledge. The reported results suggest that cross-domain transfer is beneficial in the scenarios that we investigated, lending further evidence that this strategy is a broadly effective means for increasing the efficiency of learning systems. We conclude by discussing the aspects of inductive process modeling that encourage effective transfer, by reviewing related strategies, and by describing future research plans for constraint induction and transfer learning.

In this paper, we introduce an approach for extracting constraints on process model construction. We begin by clarifying the type of knowledge produced by our method and how one may apply it. Next, we reviewthe task of inductive process modeling, which provides the required data. We then introduce a logical formalismand a computational method for acquiring scientific knowledge from candidate process models. Results suggestthat the learned constraints make sense ecologically and may provide insight into the nature of the modeled domain. We conclude the paper by discussing related and future work.

In this paper, we review the paradigm of inductive process modeling, which uses background knowledge about possible component processes to construct quantitative models of dynamical systems. We note that previous methods for this task tend to overfit the training data, which suggests ensemble learning as a likely response. However, such techniques combine models in ways that reduce comprehensibility, making their output much less accessible to domain scientists. As an alternative, we introduce a new approach that induces a set of process models from di#erent samples of the training data and uses them to guide a final search through the space of model structures. Experiments with synthetic and natural data suggest this method reduces error and decreases the chance of including unnecessary processes in the model.

Rule learning is typically used for solving classification and prediction tasks. However learning of classification rules can be adapted also to subgroup discovery. This paper shows how this can be achieved by modifying the covering algorithm and the search heuristic, performing probabilistic classification of instances, and using an appropriate measure for evaluating the results of subgroup discovery. Experimental evaluation of the CN2-SD subgroup discovery algorithm on 17 UCI data sets demonstrates substantial reduction of the number of induced rules, increased rule coverage and rule significance, as well as slight improvements in terms of the area under the ROC curve.

Meta decision trees (MDTs) are a method for combining multiple classifiers. We present an integration of the algorithm MLC4.5 for learning MDTs into the Weka data mining suite. We compare classifier ensembles combined with MDTs to bagged and boosted decision trees, and to classifier ensembles combined with other methods: voting and stacking with three different meta-level classifiers (ordinary decision trees, naive Bayes, and multi-response linear regression - MLR). Meta decision trees. Techniques for combining predictions obtained from multiple base-level classifiers can be clustered in three combining frameworks: voting (used in bagging and boosting), stacked generalization or stacking [7] and cascading. Meta decision trees (MDTs) [5] adopt the stacking framework of combining base-level classifiers. The difference between meta and ordinary decision trees (ODTs) is that MDT leaves specify which base-level classifier should be used, instead of predicting the class value directly. Th...