In recent years, symbolic regression has been of wide interest to provide an interpretable symbolic representation of potentially large data relationships. Initially circled to genetic algorithms, symbolic regression methods now include a variety of Deep Learning based alternatives. However, these methods still do not generalize well to real-world data, mainly because they hardly include domain knowledge nor consider physical relationships between variables such as known equations and units. Regarding these issues, we propose a Reinforcement-Based Grammar-Guided Symbolic Regression (RBG2-SR) method that constrains the representational space with domain-knowledge using context-free grammar as reinforcement action space. We detail a Partially-Observable Markov Decision Process (POMDP) modeling of the problem and benchmark our approach against state-of-the-art methods. We also analyze the POMDP state definition and propose a physical equation search use case on which we compare our approach to grammar-based and non-grammarbased symbolic regression methods. The experiment results show that our method is competitive against other state-of-the-art methods on the benchmarks and offers the best error-complexity trade-off, highlighting the interest of using a grammar-based method in a real-world scenario.
The goal of neural-symbolic computation is to integrate the connectionist and symbolist paradigms. Prior methods learn the neural-symbolic models using reinforcement learning (RL) approaches, which ignore the error propagation in the symbolic reasoning module and thus converge slowly with sparse rewards. In this paper, we address these issues and close the loop of neural-symbolic learning by (1) introducing the \textbf{grammar} model as a \textit{symbolic prior} to bridge neural perception and symbolic reasoning, and (2) proposing a novel \textbf{back-search} algorithm which mimics the top-dow…
Einstein Center for Neurosciences Berlin（柏林爱因斯坦神经科学中心）；Technical University Berlin（柏林工业大学）；Imperial College London（帝国理工学院）
Hierarchical Reinforcement Learning algorithms have successfully been applied to temporal credit assignment problems with sparse reward signals. However, state-of-the-art algorithms require manual specification of sub-task structures, a sample inefficient exploration phase or lack semantic interpretability. Humans, on the other hand, efficiently detect hierarchical sub-structures induced by their surroundings. It has been argued that this inference process universally applies to language, logical reasoning as well as motor control. Therefore, we propose a cognitive-inspired Reinforcement Learn…
SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校数学系）；Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校机械与航空航天工程系）；Office of Advanced Research Computing (OARC), University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校高级计算研究办公室）
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Peking University（北京大学）
Future predictions on sequence data (e.g., videos or audios) require the algorithms to capture non-Markovian and compositional properties of high-level semantics. Context-free grammars are natural choices to capture such properties, but traditional grammar parsers (e.g., Earley parser) only take symbolic sentences as inputs. In this paper, we generalize the Earley parser to parse sequence data which is neither segmented nor labeled. This generalized Earley parser integrates a grammar parser with a classifier to find the optimal segmentation and labels, and makes top-down future predictions. Ex…
We introduce NSEdit (neural-symbolic edit), a novel Transformer-based code repair method. Given only the source code that contains bugs, NSEdit predicts an editing sequence that can fix the bugs. The edit grammar is formulated as a regular language, and the Transformer uses it as a neural-symbolic scripting interface to generate editing programs. We modify the Transformer and add a pointer network to select the edit locations. An ensemble of rerankers are trained to re-rank the editing sequences generated by beam search. We fine-tune the rerankers on the validation set to reduce over-fitting. …
University of Ljubljana（卢布尔雅那大学）；Jožef Stefan Institute（约瑟夫·斯特凡研究所）
Probabilistic context-free grammars have a long-term record of use as generative models in machine learning and symbolic regression. When used for symbolic regression, they generate algebraic expressions. We define the latter as equivalence classes of strings derived by grammar and address the problem of calculating the probability of deriving a given expression with a given grammar. We show that the problem is undecidable in general. We then present specific grammars for generating linear, polynomial, and rational expressions, where algorithms for calculating the probability of a given expres…
In this paper, I describe several approaches to automatic or semi-automatic development of symbolic rules for grammar checkers from the information contained in corpora. The rules obtained this way are an important addition to manually-created rules that seem to dominate in rule-based checkers. However, the manual process of creation of rules is costly, time-consuming and error-prone. It seems therefore advisable to use machine-learning algorithms to create the rules automatically or semi-automatically. The results obtained seem to corroborate my initial hypothesis that symbolic machine learni…
Neuro-Symbolic AI for Analytical Solutions of Differential Equations
Seminar for Applied Mathematics, ETH Zurich, Switzerland（应用数学研讨会，苏黎世联邦理工学院，瑞士）；ETH AI Center, Zurich, Switzerland（苏黎世联邦理工学院人工智能中心，瑞士）；IBM Research Europe, Zurich, Switzerland（IBM欧洲研究院，苏黎世，瑞士）；Institute for Atmospheric and Climate Science, ETH Zurich, Switzerland（大气与气候科学研究所，苏黎世联邦理工学院，瑞士）；Swiss Data Science Center, ETH Zurich, Switzerland（瑞士数据科学中心，苏黎世联邦理工学院，瑞士）
APP Centre for Artificial Intelligence Research（APP人工智能研究中心）；BITS Pilani（毕拉尼理工学院）；TCS Innovation Labs（塔塔咨询服务公司创新实验室）
Our interest is in scientific problems with the following characteristics: (1) Data are naturally represented as graphs; (2) The amount of data available is typically small; and (3) There is significant domain-knowledge, usually expressed in some symbolic form. These kinds of problems have been addressed effectively in the past by Inductive Logic Programming (ILP), by virtue of 2 important characteristics: (a) The use of a representation language that easily captures the relation encoded in graph-structured data, and (b) The inclusion of prior information encoded as domain-specific relations, …
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；Boston University（波士顿大学）
Flexible neural sequence models outperform grammar- and automaton-based counterparts on a variety of tasks. However, neural models perform poorly in settings requiring compositional generalization beyond the training data -- particularly to rare or unseen subsequences. Past work has found symbolic scaffolding (e.g. grammars or automata) essential in these settings. We describe R&R, a learned data augmentation scheme that enables a large category of compositional generalizations without appeal to latent symbolic structure. R&R has two components: recombination of original training examples via …
University of Applied Sciences Upper Austria（上奥地利应用科技大学）；Johannes Kepler University（约翰内斯·开普勒大学）
Symbolic regression is a powerful system identification technique in industrial scenarios where no prior knowledge on model structure is available. Such scenarios often require specific model properties such as interpretability, robustness, trustworthiness and plausibility, that are not easily achievable using standard approaches like genetic programming for symbolic regression. In this chapter we introduce a deterministic symbolic regression algorithm specifically designed to address these issues. The algorithm uses a context-free grammar to produce models that are parameterized by a non-line…
University of Applied Sciences Upper Austria（上奥地利应用科技大学）；Johannes Kepler University（约翰内斯·开普勒大学）
In this chapter we take a closer look at the distribution of symbolic regression models generated by genetic programming in the search space. The motivation for this work is to improve the search for well-fitting symbolic regression models by using information about the similarity of models that can be precomputed independently from the target function. For our analysis, we use a restricted grammar for uni-variate symbolic regression models and generate all possible models up to a fixed length limit. We identify unique models and cluster them based on phenotypic as well as genotypic similarity…
University of Eindhoven（埃因霍温大学）；Ludwig-Maximilians-University（路德维希-马克西米利安大学）；LIACS, Leiden University（莱顿大学高级计算机科学研究所）
Hyperparameter optimization in machine learning (ML) deals with the problem of empirically learning an optimal algorithm configuration from data, usually formulated as a black-box optimization problem. In this work, we propose a zero-shot method to meta-learn symbolic default hyperparameter configurations that are expressed in terms of the properties of the dataset. This enables a much faster, but still data-dependent, configuration of the ML algorithm, compared to standard hyperparameter optimization approaches. In the past, symbolic and static default values have usually been obtained as han…
Physics-informed Neural Networks (PINNs) have been widely used to obtain accurate neural surrogates for a system of Partial Differential Equations (PDE). One of the major limitations of PINNs is that the neural solutions are challenging to interpret, and are often treated as black-box solvers. While Symbolic Regression (SR) has been studied extensively, very few works exist which generate analytical expressions to directly perform SR for a system of PDEs. In this work, we introduce an end-to-end framework for obtaining mathematical expressions for solutions of PDEs. We use a trained PINN to ge…
Vanderbilt University Medical Center（范德堡大学医学中心）；Clare Hall, University of Cambridge（剑桥大学克莱尔学堂）
We develop a symbolic regression framework for extracting the governing mathematical expressions from observed data. The evolutionary approach, faiGP, is designed to leverage the properties of a function algebra that have been encoded into a grammar, providing a theoretical guarantee of universal approximation and a way to minimize bloat. In this framework, the choice of operators of the grammar may be informed by a physical theory or symmetry considerations. Since there is currently no theory that can derive the 'constants of nature', an empirical investigation on extracting these coefficient…
Recurrent neural networks are powerful models for sequential data, able to represent complex dependencies in the sequence that simpler models such as hidden Markov models cannot handle. Yet they are notoriously hard to train. Here we introduce a training procedure using a gradient ascent in a Riemannian metric: this produces an algorithm independent from design choices such as the encoding of parameters and unit activities. This metric gradient ascent is designed to have an algorithmic cost close to backpropagation through time for sparsely connected networks. We use this procedure on gated le…
In the recent past, there has been a growing interest in Neural-Symbolic Integration frameworks, i.e., hybrid systems that integrate connectionist and symbolic approaches to obtain the best of both worlds. In a previous work, we proposed KENN (Knowledge Enhanced Neural Networks), a Neural-Symbolic architecture that injects prior logical knowledge into a neural network by adding a new final layer which modifies the initial predictions accordingly to the knowledge. Among the advantages of this strategy, there is the inclusion of clause weights, learnable parameters that represent the strength of…
In the recent past, there has been a growing interest in Neural-Symbolic Integration frameworks, i.e., hybrid systems that integrate connectionist and symbolic approaches to obtain the best of both worlds. In this work we focus on a specific method, KENN (Knowledge Enhanced Neural Networks), a Neural-Symbolic architecture that injects prior logical knowledge into a neural network by adding on its top a residual layer that modifies the initial predictions accordingly to the knowledge. Among the advantages of this strategy, there is the inclusion of clause weights, learnable parameters that repr…
Josef Ressel Center for Symbolic Regression（约瑟夫·雷塞尔符号回归中心）；Heuristic and Evolutionary Algorithms Laboratory（启发式与进化算法实验室）；University of Applied Sciences Upper Austria（上奥地利应用科学大学）
Shape-constrained symbolic regression (SCSR) allows to include prior knowledge into data-based modeling. This inclusion allows to ensure that certain expected behavior is better reflected by the resulting models. The expected behavior is defined via constraints, which refer to the function form e.g. monotonicity, concavity, convexity or the models image boundaries. In addition to the advantage of obtaining more robust and reliable models due to defining constraints over the functions shape, the use of SCSR allows to find models which are more robust to noise and have a better extrapolation beh…
This document develops general concepts useful for extracting knowledge embedded in large graphs or datasets that have pair-wise relationships, such as cause-effect-type relations. Almost no underlying assumptions are made, other than that the data can be presented in terms of pair-wise relationships between objects/events. This assumption is used to mine for patterns in the dataset, defining a reduced graph or dataset that boils-down or concentrates information into a more compact form. The resulting extracted structure or set of patterns are manifestly symbolic in nature, as they capture and…
