Symbolic Regression for Shared Expressions: Introducing Partial Parameter Sharing

Symbolic Regression aims to find symbolic expressions that describe datasets. Due to better interpretability, it is a machine learning paradigm particularly powerful for scientific discovery. In recent years, several works have expanded the concept to allow the description of similar phenomena using a single expression with varying sets of parameters, thereby introducing categorical variables. Some previous works allow only "non-shared" (category-value-specific) parameters, and others also incorporate "shared" (category-value-agnostic) parameters. We expand upon those efforts by considering multiple categorical variables, and introducing intermediate levels of parameter sharing. With two categorical variables, an intermediate level of parameter sharing emerges, i.e., parameters which are shared across either category but change across the other. The new approach potentially decreases the number of parameters, while revealing additional information about the problem. Using a synthetic, fitting-only example, we test the limits of this setup in terms of data requirement reduction and transfer learning. As a real-world symbolic regression example, we demonstrate the benefits of the proposed approach on an astrophysics dataset used in a previous study, which considered only one categorical variable. We achieve a similar fit quality but require significantly fewer individual parameters, and extract additional information about the problem.

Geometry Denoising with Preferred Normal Vectors

We introduce a new paradigm for geometry denoising using prior knowledge about the surface normal vector. This prior knowledge comes in the form of a set of preferred normal vectors, which we refer to as label vectors. A segmentation problem is naturally embedded in the denoising process. The segmentation is based on the similarity of the normal vector to the elements of the set of label vectors. Regularization is achieved by a total variation term. We formulate a split Bregman (ADMM) approach to solve the resulting optimization problem. The vertex update step is based on second-order shape calculus.

Total Generalized Variation of the Normal Vector Field and Applications to Mesh Denoising

We propose a novel formulation for the second-order total generalized variation (TGV) of the normal vector on an oriented, triangular mesh embedded in $\mathbb{R}^3$. The normal vector is considered as a manifold-valued function, taking values on the unit sphere. Our formulation extends previous discrete TGV models for piecewise constant scalar data that utilize a Raviart-Thomas function space. To exctend this formulation to the manifold setting, a tailor-made tangential Raviart-Thomas type finite element space is constructed in this work. The new regularizer is compared to existing methods in mesh denoising experiments.

Two Models for Surface Segmentation using the Total Variation of the Normal Vector

We consider the problem of surface segmentation, where the goal is to partition a surface represented by a triangular mesh. The segmentation is based on the similarity of the normal vector field to a given set of label vectors. We propose a variational approach and compare two different regularizers, both based on a total variation measure. The first regularizer penalizes the total variation of the assignment function directly, while the second regularizer penalizes the total variation in the label space. In order to solve the resulting optimization problems, we use variations of the split Bregman (ADMM) iteration adapted to the problem at hand. While computationally more expensive, the second regularizer yields better results in our experiments, in particular it removes noise more reliably in regions of constant curvature.

Shape Constraints in Symbolic Regression using Penalized Least Squares

We study the addition of shape constraints and their consideration during the parameter estimation step of symbolic regression (SR). Shape constraints serve as a means to introduce prior knowledge about the shape of the otherwise unknown model function into SR. Unlike previous works that have explored shape constraints in SR, we propose minimizing shape constraint violations during parameter estimation using gradient-based numerical optimization. We test three algorithm variants to evaluate their performance in identifying three symbolic expressions from a synthetically generated data set. This paper examines two benchmark scenarios: one with varying noise levels and another with reduced amounts of training data. The results indicate that incorporating shape constraints into the expression search is particularly beneficial when data is scarce. Compared to using shape constraints only in the selection process, our approach of minimizing violations during parameter estimation shows a statistically significant benefit in some of our test cases, without being significantly worse in any instance.

Unit-Aware Genetic Programming for the Development of Empirical Equations

When developing empirical equations, domain experts require these to be accurate and adhere to physical laws. Often, constants with unknown units need to be discovered alongside the equations. Traditional unit-aware genetic programming (GP) approaches cannot be used when unknown constants with undetermined units are included. This paper presents a method for dimensional analysis that propagates unknown units as ''jokers'' and returns the magnitude of unit violations. We propose three methods, namely evolutive culling, a repair mechanism, and a multi-objective approach, to integrate the dimensional analysis in the GP algorithm. Experiments on datasets with ground truth demonstrate comparable performance of evolutive culling and the multi-objective approach to a baseline without dimensional analysis. Extensive analysis of the results on datasets without ground truth reveals that the unit-aware algorithms make only low sacrifices in accuracy, while producing unit-adherent solutions. Overall, we presented a promising novel approach for developing unit-adherent empirical equations.

Adaptive Step Sizes for Preconditioned Stochastic Gradient Descent

This paper proposes a novel approach to adaptive step sizes in stochastic gradient descent (SGD) by utilizing quantities that we have identified as numerically traceable -- the Lipschitz constant for gradients and a concept of the local variance in search directions. Our findings yield a nearly hyperparameter-free algorithm for stochastic optimization, which has provable convergence properties when applied to quadratic problems and exhibits truly problem adaptive behavior on classical image classification tasks. Our framework enables the potential inclusion of a preconditioner, thereby enabling the implementation of adaptive step sizes for stochastic second-order optimization methods.

Sensitivity-Based Layer Insertion for Residual and Feedforward Neural Networks

The training of neural networks requires tedious and often manual tuning of the network architecture. We propose a systematic method to insert new layers during the training process, which eliminates the need to choose a fixed network size before training. Our technique borrows techniques from constrained optimization and is based on first-order sensitivity information of the objective with respect to the virtual parameters that additional layers, if inserted, would offer. We consider fully connected feedforward networks with selected activation functions as well as residual neural networks. In numerical experiments, the proposed sensitivity-based layer insertion technique exhibits improved training decay, compared to not inserting the layer. Furthermore, the computational effort is reduced in comparison to inserting the layer from the beginning. The code is available at \url{https://github.com/LeonieKreis/layer_insertion_sensitivity_based}.

Frobenius-Type Norms and Inner Products of Matrices and Linear Maps with Applications to Neural Network Training

The Frobenius norm is a frequent choice of norm for matrices. In particular, the underlying Frobenius inner product is typically used to evaluate the gradient of an objective with respect to matrix variable, such as those occuring in the training of neural networks. We provide a broader view on the Frobenius norm and inner product for linear maps or matrices, and establish their dependence on inner products in the domain and co-domain spaces. This shows that the classical Frobenius norm is merely one special element of a family of more general Frobenius-type norms. The significant extra freedom furnished by this realization can be used, among other things, to precondition neural network training.

Introducing Thermodynamics-Informed Symbolic Regression -- A Tool for Thermodynamic Equations of State Development

Thermodynamic equations of state (EOS) are essential for many industries as well as in academia. Even leaving aside the expensive and extensive measurement campaigns required for the data acquisition, the development of EOS is an intensely time-consuming process, which does often still heavily rely on expert knowledge and iterative fine-tuning. To improve upon and accelerate the EOS development process, we introduce thermodynamics-informed symbolic regression (TiSR), a symbolic regression (SR) tool aimed at thermodynamic EOS modeling. TiSR is already a capable SR tool, which was used in the research of https://doi.org/10.1007/s10765-023-03197-z. It aims to combine an SR base with the extensions required to work with often strongly scattered experimental data, different residual pre- and post-processing options, and additional features required to consider thermodynamic EOS development. Although TiSR is not ready for end users yet, this paper is intended to report on its current state, showcase the progress, and discuss (distant and not so distant) future directions. TiSR is available at https://github.com/scoop-group/TiSR and can be cited as https://doi.org/10.5281/zenodo.8317547.