Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the quantitative approximation theorem for solution operators of nonlinear PDEs. Our results show that neural operators can efficiently approximate these solution operators without the exponential growth in model complexity, thus strengthening the theoretical foundation of neural operators. A key insight in our proof is to transfer PDEs into the corresponding integral equations via Duahamel's principle, and to leverage the similarity between neural operators and Picard’s iteration—a classical algorithm for solving PDEs. This approach is potentially generalizable beyond parabolic PDEs to a class of PDEs which can be solved by Picard's iteration.

Recurrent neural networks (RNNs) are a class of neural networks used in sequential tasks. However, in general, RNNs have a large number of parameters and involve enormous computational costs by repeating the recurrent structures in many time steps. As a method to overcome this difficulty, RNN pruning has attracted increasing attention in recent years, and it brings us benefits in terms of the reduction of computational cost as the time step progresses. However, most existing methods of RNN pruning are heuristic. The purpose of this paper is to study the theoretical scheme for RNN pruning method. We propose an appropriate pruning algorithm for RNNs inspired by "spectral pruning", and provide the generalization error bounds for compressed RNNs. We also provide numerical experiments to demonstrate our theoretical results and show the effectiveness of our pruning method compared with the existing methods.

Up to now, different kinds of musical performance robots (MPRs) and robotic musicians (RMs) have been developed. MPRs are designed to closely reproduce the motor skills displayed by humans in order to play musical instruments. From this approach, MPRs are used as benchmarks to study the human motor control from an engineering point of view and to understand better the human-robot interaction from a musical point of view. In contrast, RMs are conceived as automated mechanisms designed to create new ways of musical expression from a musical engineering point of view. Our research, at Waseda University, has been focused on developing an anthropomorphic flutist robot. Our research aims in studying the human motor control from an engineering point of view, understanding the ways to facilitate the human-robot interaction and proposing new applications for humanoid robot in musical terms. As a result of our research, the Waseda Flutist Robot No.4 Refined IV (WF-4RIV) has been developed. In a previous research, we focused on improving the mechanical system in order to enhance the clarity of the sound. However, we require performing further improvements to the control system in order to enable the robot to autonomously improve the quality of the sound during the flute performance. For this purpose, we proposed to implement an auditory feedback control system on the flutist robot. The proposed system is composed by a music expressive generator, feed-forward air pressure control system and a pitch evaluation module. From the experimental results with the WF-4RIV, we could confirm the improvements on the flute performance.

Up to now, different kinds of musical performance robots (MPRs) have been developed. MPRs are designed to closely reproduce the required motor skills displayed by humans in order to play musical instruments. Our research, at Waseda University, has been focused on developing an anthropomorphic flutist robot. In this year, we have succeeded in developing the Waseda Flutist Robot No.4 Refined IV (WF-4RIV); which has improved the design of the mouth mechanism and reproduced the anatomy of the human lips better. In addition, the control system was improved to enable the robot to autonomously improve the quality of the sound during performance. For this purpose, we proposed to implement a new performance control system. From the experimental results, we confirm the improvements on the flute sound produced by the robot.

In this paper, the methodology for automatically generating an expressive performance on the anthropomorphic flutist robot is detailed. A feed-forward network trained with the error back-propagation algorithm was implemented to model the performance's expressiveness of a professional flutist. In particular, the note duration and vibrato were considered as performance rules (sources of variation) to enhance the robot's performance expressiveness. From the mechanical point of view, the vibrato and lung systems were re-designed to effectively control the proposed music performance rules. An experimental setup was proposed to verify the effectiveness of generating a new score with expressiveness from a model created based on the performance of a professional flutist. As a result, the flutist robot was able of automatically producing an expressive performance similar to the human one from a nominal score.

Our research on the anthropomorphic flutist robot has been focused on understanding the human motor control, enabling the communication between human and robots, proposing novel applications for humanoid robots and introducing novel ways of musical expression. In this video, we are presenting the musical skills of the Waseda Flutist Robot. In particular, we are presenting the WF-4RIV; which is able of performing musical performances requiring basic as well as extended technical skills. Thanks to the most recent mechanical improvements, the WF-4RIV is able of performing musical scores that are commonly part of the repertoire of intermediate level players. In this video, we are presenting the WF-4RIV performing the Flight of the Bumble Bee, Le Cygne and Trois Duos de Mendelssohn et Lachner.

As a result from our research, the Waseda Flutist Robot is able of playing the flute nearly similar to an intermediate level flutist. In order to enhance the expressiveness of its performance, we are focusing our research on improving the mechanical design of the simulated organs as well as implementing automated algorithms for the generation of expressive music performance. In a previous research, we have developed a human-like vocal cord to improve the production of vibrato. In this paper, we are presenting the newest version of the flutist robot; where the lips, oral cavity and tonguing mechanisms were improved. Such improvements were proposed to effectively control the attack time and double tonguing. The attack time is useful to produce clear sounds and the double tonguing is an important articulation that helps players to produce shaped notes and smooth transitions between notes. The lips mechanism is composed by 3-DOFs which enables the accurate control of the air stream parameters (width, thickness and angle). The lips of the robot were designed more human-like by using a thermoplastic rubber (septon). The oral cavity was designed similar to the human one also made by septon. Inside the oral cavity, an improved tonguing mechanism was designed (1-DOF) to reproduce better the double tonguing technique. A set of experiments were performed to analyze the improvements on the dynamical properties of the sound while playing the flute.

The research on the anthropomorphic flutist robot, at Waseda University, has focused on clarifying the human motor control while playing the flute, proposing novel applications for humanoid robots and enabling the communication with humans at the emotional level. As a result of our research, the flutist robot is able of nearly reproducing the basic technical skills required to play the flute. Furthermore, some of the extended technical skills have been roughly simulated by the robot. However, in order to enhance the expressiveness of the robot's performance, still further improvements are required. In particular, in this paper, we focus our research on understanding better how to enhance the expressiveness of the flute performance by studying in more detail the vibrato, which is believe to be one of the most important ways of expressing feelings/ideas while performing the flute. For this reason, the newest version of the flutist robot, the Waseda Flutist Robot No.4 Refined III has been developed by focusing on improving the design of the lung and designing of a human-like vocal cord mechanism, which are believed to have a close relation in producing vibrato. The details of the new mechanism are given and experiments were conducted to understand the effect of the movement of diaphragm and glottis while producing vibrato.