Evaluation of basic mathematical abilities of neural networks

openIn human cognition, when advanced mathematical abilities reach a certain level, basic numerical skills, such as number sense and elementary calculation, are typically well-developed. In this thesis we investigate whether state-of-the-art artificial neural network models exhibit a similar trend. Indeed, much research has pointed out that large-scale language models (such as ChatGPT) possess exceptional high-level mathematical abilities, but their elementary numeracy skills have often been overlooked. This dissertation focuses on the foundational mathematical abilities of GPT-3.5 (from which ChatGPT was developed), its newest version GPT-4 and six other multi-modal deep learning models. Taking into account the unique characteristics of different neural network models, standardized tests and self-developed tasks were employed to explore the mathematical abilities of these eight models. The findings indicate that GPT-3.5 and GPT-4 are indeed able to exhibit complex mathematical competencies, though basic numeracy skills are not always fully developed (especially in GPT-3.5). In contrast, the six multi-modal models still need to make progress in improving their numerosity perception and number sense to unlock more advanced mathematical abilities.In human cognition, when advanced mathematical abilities reach a certain level, basic numerical skills, such as number sense and elementary calculation, are typically well-developed. In this thesis we investigate whether state-of-the-art artificial neural network models exhibit a similar trend. Indeed, much research has pointed out that large-scale language models (such as ChatGPT) possess exceptional high-level mathematical abilities, but their elementary numeracy skills have often been overlooked. This dissertation focuses on the foundational mathematical abilities of GPT-3.5 (from which ChatGPT was developed), its newest version GPT-4 and six other multi-modal deep learning models. Taking into account the unique characteristics of different neural network models, standardized tests and self-developed tasks were employed to explore the mathematical abilities of these eight models. The findings indicate that GPT-3.5 and GPT-4 are indeed able to exhibit complex mathematical competencies, though basic numeracy skills are not always fully developed (especially in GPT-3.5). In contrast, the six multi-modal models still need to make progress in improving their numerosity perception and number sense to unlock more advanced mathematical abilities

Virtual reality based creation of concept model designs for CAD systems

This work introduces a novel method to overcome most of the drawbacks in traditional methods for creating design models. The main innovation is the use of virtual tools to simulate the natural physical environment in which freeform. Design models are created by experienced designers. Namely, the model is created in a virtual environment by carving a work piece with tools that simulate NC milling cutters. Algorithms have been developed to support the approach, in which the design model is created in a Virtual Reality (VR) environment and selection and manipulation of tools can be performed in the virtual space. The desianer\u27s hand movements generate the tool trajectories and they are obtained by recording the position and orientation of a hand mounted motion tracker. Swept volumes of virtual tools are generated from the geometry of the tool and its trajectories. Then Boolean operations are performed on the swept volumes and the initial virtual stock (work piece) to create the design model. Algorithms have been developed as a part of this work to integrate the VR environment with a commercial CAD/CAM system in order to demonstrate the practical applications of the research results. The integrated system provides a much more efficient and easy-to-implement process of freeform model creation than employed in current CAD/CAM software. It could prove to be the prototype for the next-generation CAD/CAM system

From 3D Models to 3D Prints: an Overview of the Processing Pipeline

Due to the wide diffusion of 3D printing technologies, geometric algorithms for Additive Manufacturing are being invented at an impressive speed. Each single step, in particular along the Process Planning pipeline, can now count on dozens of methods that prepare the 3D model for fabrication, while analysing and optimizing geometry and machine instructions for various objectives. This report provides a classification of this huge state of the art, and elicits the relation between each single algorithm and a list of desirable objectives during Process Planning. The objectives themselves are listed and discussed, along with possible needs for tradeoffs. Additive Manufacturing technologies are broadly categorized to explicitly relate classes of devices and supported features. Finally, this report offers an analysis of the state of the art while discussing open and challenging problems from both an academic and an industrial perspective.Comment: European Union (EU); Horizon 2020; H2020-FoF-2015; RIA - Research and Innovation action; Grant agreement N. 68044

Improving precision of material extrusion 3D printing by in-situ monitoring and predicting 3D geometric deviation using Conditional Adversarial Networks

The field of additive manufacturing, especially 3D printing, has gained growing attention in the research and commercial sectors in recent years. Notwithstanding that the capabilities of 3D printing have moved on to enhanced resolution, higher deposition rate, and a wide variety of materials, the crucial challenge of verifying that the component manufactured is within the dimensional tolerance as designed continues to exist. Material extrusion 3D printing has long been established for rapid prototyping and functional testing in many research and industry fields. However, its inconsistency and intrinsic defects (surface roughness and geometric inaccuracies) hinder its application in several areas, most notably “certify-as-you- build” small-batch prototyping and large-batch production.In this study, we present an approach to reduce both inconsistency and the 3D geometric inaccuracies of products fabricated by material extrusion.1. This work developed and demonstrated an approach for layer-by-layer mapping of 3D printed parts, which can be used for validation of printed models and in situ adjustment of print parameters. This in situ metrology system scans each layer at the time of printing, providing a 3D model of the as-printed part. A high-speed optical scanning system was integrated with a Material Extrusion type 3D printer to achieve in situ monitoring of dimensional inaccuracies during printing, which leaves the door open to implement a closed-loop feedback system to compensate geometric errors during printing in the future and fabricate “certify-as-you-build” products.2. This work trained machine learning algorithms with data from this scanning system and predicted 3D geometric inaccuracies in new designs. Eight Conditional Adversarial Networks (CAN) machine learning models were trained on a limited number of scanned profile images of different layers, consisting of less than 50 actual images and 50 generated images, to predict the 3D geometric deviations of freeform shapes. The generated images were produced by randomly combining and cropping the actual images without any distortion. These CAN models produced predictions where at least 44.4%, 87.6%, 99.2% of data were within �0.05 mm, �0.10 mm, �0.15 mm of the actual measured value, respectively.3. This work developed an Iterative Forward approach to redesign the Computer-Aided- Design model by reverse engineering using the trained machine learning models, allowing for compensation of print imperfection at the design stage, in advance of the first printing. The compensation algorithms with eight different sets of different parameters were evaluated. It has been proven that the Iterative Forward approach improved the geometric deviation of the predicted profiles by making compensation to the CAD model

Researches in non-associative algebra

I have frequently been asked by biologists for mathematical help in connection with their problems. I was working on one such problem when an algebraist, observing my work without knowing what it was about, remarked that I was apparently using hypercomplex numbers. I was considering a certain type of inheritance specified by formulae which could be regarded as forming the multiplication table of a non -associative linear algebra; and my calculations could be regarded as manipulations of hypercomplex numbers in this algebra, or in another algebra derived from it by a process which I later called "duplication;I then realised that there are many such "genetic algebras ", representing different types of inheritance. They are in all cases non-associative as regards multiplication, though they can always be taken to be commutative. I found that a large class of genetic algebras (viz. those for "symmetrical inheritance" as defined in Paper VI, p. 2) possessed certain distinctive properties which seemed worthy of investigation for their own sake, and also for the sake of possible exploitation in genetics.Part Three, the main part of this thesis, consists of four papers in which this investigation is given - or rather is begun, for there are a good many problems left untackled.Part One consists of four papers (one written in collaboration with Dr A. Erdélyi) on some purely combinatory problems of non - associative algebra, suggested by the notations which I employed for products and powers in the genetic algebras. The combinatory per t 0.4-01.5 rt. theory is continued in theAconcluding postscipt which follows Paper X.Part Two shows how genetic algebras arise and are manipulate The multiplication table of a genetic algebra, the multiplication of hypercomplex numbers, and the above mentioned process of duplication, are simply a translation into symbols of the relevant essentials in the processes; of inheritance; and the symbolism as a whole is a convenient shorthand for reckoning with combinations and statistical distributions of genetic types, enabling one to dispense with some of the verbal arguments and the chessboard diagrams commonly used for the same purpose. In paper VI the treatment is made as general as possible with the object of showirg the relationship between different genetic algebras and something of their structure; and the concepts to be discussed in Part Three are here defined. In Paper V, which was published later but mostly written earlier than VI, the explanation is given in very much simpler mathematical language (for it was intended to be read by geneticists), and with more attention to practical applications. It can be explained very simply why multiplication in the genetic algebras is non- associative, that is to say(AB) C ≠ A (BC)This statement is interpreted:- "If the offspring of A and B mates with C, the probability distribution of genetic types in the progeny will not be the same as if A mates with the offspring of B and C."My symbolism was not essentially new: the novelty lay in is interpretationlin terms of hypercomplex numbers. In fact it could be said that genetic algebras had been used by geneticists in a primitive way for quite a long time without having been recognised. explicitly. Their explicit recognition is I believe more than a mere change of notation. Apart from greater brevity achieved in some applications, general theorems on linear algebras can be applied; transformations can be used which are quite meaningless genetically but which lead to genetically significant conclusions; and the use of an index notation and summation convention reduces the symbolism to manageable proportions when, with inheritance involving many genes, it threatens to become too heavy to handle.Biological considerations were thus the root of these researches, and I intend to return to the genetical applications later; for I believe that genetic algebras may throw light on some deeper problems of genetics. I cannot at present give solid justification for this belief in the sense of having successfully tackled problems otherwise unsolved, and I therefore wish that this thesis may not be judged as a finished achievement in biological investigation; but may be judged primarily as a contribution to algebra, suggested by biological problems, and perhaps having possibilities of application beyond the simple ones so far demonstrated

Generation of office buildings in large scale virtual worlds

Virtual worlds are used in many different areas, from military training simulations to massive multiplayer online role-playing games. In the past, the sizes of these worlds was limited by the power of the computers that ran them as well as the man-hours needed to draw them. However, as computers have become more powerful, the limiting fctor has become the man-hours needed to manually draw every object in such a world. So there is now a need for large scale, traversable, dynamic, algorithmically generated virtual worlds. For these worlds to be realistic, cities need to be generated, and for these cities to be relistic, they must have commercial office buildings (skyscrapers, office parks, etc.). Previous research in this area has been solely on generating the outsides of commercial buildings, with no focus on the inside features of the buildings. This research aims to generate both the insides and the outsides of commercial office buildings, with the dual goals of realism and usability

Point-based mathematics for computer-aided manufacture

This thesis demonstrates the feasibility of machining high quality sculptured surfaces directly from a point-based definition. The work is founded on the strategy of using a sparse set of points to characterise shape although it is assumed that an appropriately dense definition can be generated by the use of some unspecified high quality interpolation algorithm. This is in contrast to the conventional CAD/CAM approach where explicit parametric expressions are used to describe the part. The research is founded on the Inverse Offset Method (IOM) proposed by Kishinami; the algorithm is chosen because it possesses a number of desirable properties, most notably its versatility and robustness. The first fundamental contribution is an error analysis of the IOM that has not been published before, the analysis is dependent on the surface and cutter path point spacing, the tool radius and the local surface curvature. The accuracy of the error analysis is corroborated by the machining and measuring of a physical part. Furthermore it is established that the quality of the finished part produced by the IOM compares favourably with that produced by a commercial package for similar tolerances. The principal research achievement is the optimisation of the IOM to exploit the coherence of data ordered into sections. This results in the IOM generating cutter paths in a time period comparable to that of the commercial package without a reduction in the quality of the finished part. The last contribution made in this thesis is a report on the issues concerning the machining of point definitions derived from multi-surfaces. The work presented in this thesis offers an alternative strategy to the design and manufacture of free-form surfaces. The main benefits of adopting this strategy are gained because it removes the need to generate a parametric surface definition

Mathematical techniques for shape modelling in computer graphics: A distance-based approach.

This research is concerned with shape modelling in computer graphics. The dissertation provides a review of the main research topics and developments in shape modelling and discusses current visualisation techniques required for the display of the models produced. In computer graphics surfaces are normally defined using analytic functions. Geometry however, supplies many shapes without providing their analytic descriptions. These are defined implicitly through fundamental relationships between primitive geometrical objects. Transferring this approach in computer graphics, opens new directions in shape modelling by enabling the definition of new objects or supplying a rigorous alternative to analytical definitions of objects with complex analytical descriptions. We review, in this dissertation, relevant works in the area of implicit modelling. Based on our observations on the shortcomings of these works, we develop an implicit modelling approach which draws on a seminal technique in this area: the distance based object definition. We investigate the principles, potential and applications of this technique both in conceptual terms (modelling aspects) and on technical merit (visualisation issues). This is the context of this PhD research. The conceptual and technological frameworks developed are presented in terms of a comprehensive investigation of an object's constituent primitives and modelling constraints on the one hand, and software visualisation platforms on the other. Finally, we adopt a critical perspective of our work to discuss possible directions for further improvements and exploitation for the modelling approach we have developed