Erzeugunsgrad, VC-Dimension and Neural Networks with rational activation function

The notion of Erzeugungsgrad was introduced by Joos Heintz in 1983 to bound the number of non-empty cells occurring after a process of quantifier elimination. We extend this notion and the combinatorial bounds of Theorem 2 in Heintz (1983) using the degree for constructible sets defined in Pardo-Sebasti\'an (2022). We show that the Erzeugungsgrad is the key ingredient to connect affine Intersection Theory over algebraically closed fields and the VC-Theory of Computational Learning Theory for families of classifiers given by parameterized families of constructible sets. In particular, we prove that the VC-dimension and the Krull dimension are linearly related up to logarithmic factors based on Intersection Theory. Using this relation, we study the density of correct test sequences in evasive varieties. We apply these ideas to analyze parameterized families of neural networks with rational activation function.

Dynamics of Insect Paraintelligence: How a Mindless Colony of Ants Meaningfully Moves a Beetle

In this work, a new concept called Vector Dissipation of Randomness (VDR) is developed and formalized. It describes the mechanism by which complex multicomponent systems transition from chaos to order through the filtering of random directions, accumulation of information in the environment, and self-organization of agents. VDR explains how individual random strategies can evolve into collective goaldirected behavior, leading to the emergence of an ordered structure without centralized control. To test the proposed model, a numerical simulation of the "ant and beetle" system was conducted, in which agents (ants) randomly choose movement directions, but through feedback mechanisms and filtering of weak strategies, they form a single coordinated vector of the beetles movement. VDR is a universal mechanism applicable to a wide range of self-organizing systems, including biological populations, decentralized technological networks, sociological processes, and artificial intelligence algorithms. For the first time, an equation of the normalized emergence function in the processing of vector dissipation of randomness in the Ant and Beetle system has been formulated. The concept of paraintelligence was introduced for the first time. Insect paraintelligence is interpreted as a rational functionality that is close to or equivalent to intelligent activity in the absence of reflexive consciousness and selfawareness.

REAct: Rational Exponential Activation for Better Learning and Generalization in PINNs

Physics-Informed Neural Networks (PINNs) offer a promising approach to simulating physical systems. Still, their application is limited by optimization challenges, mainly due to the lack of activation functions that generalize well across several physical systems. Existing activation functions often lack such flexibility and generalization power. To address this issue, we introduce Rational Exponential Activation (REAct), a generalized form of tanh consisting of four learnable shape parameters. Experiments show that REAct outperforms many standard and benchmark activations, achieving an MSE three orders of magnitude lower than tanh on heat problems and generalizing well to finer grids and points beyond the training domain. It also excels at function approximation tasks and improves noise rejection in inverse problems, leading to more accurate parameter estimates across varying noise levels.

Generating $π$-Functional Molecules Using STGG+ with Active Learning

Generating novel molecules with out-of-distribution properties is a major challenge in molecular discovery. While supervised learning methods generate high-quality molecules similar to those in a dataset, they struggle to generalize to out-of-distribution properties. Reinforcement learning can explore new chemical spaces but often conducts 'reward-hacking' and generates non-synthesizable molecules. In this work, we address this problem by integrating a state-of-the-art supervised learning method, STGG+, in an active learning loop. Our approach iteratively generates, evaluates, and fine-tunes STGG+ to continuously expand its knowledge. We denote this approach STGG+AL. We apply STGG+AL to the design of organic $\pi$-functional materials, specifically two challenging tasks: 1) generating highly absorptive molecules characterized by high oscillator strength and 2) designing absorptive molecules with reasonable oscillator strength in the near-infrared (NIR) range. The generated molecules are validated and rationalized in-silico with time-dependent density functional theory. Our results demonstrate that our method is highly effective in generating novel molecules with high oscillator strength, contrary to existing methods such as reinforcement learning (RL) methods. We open-source our active-learning code along with our Conjugated-xTB dataset containing 2.9 million $\pi$-conjugated molecules and the function for approximating the oscillator strength and absorption wavelength (based on sTDA-xTB).

Alleviating Overfitting in Transformation-Interaction-Rational Symbolic Regression with Multi-Objective Optimization

The Transformation-Interaction-Rational is a representation for symbolic regression that limits the search space of functions to the ratio of two nonlinear functions each one defined as the linear regression of transformed variables. This representation has the main objective to bias the search towards simpler expressions while keeping the approximation power of standard approaches. The performance of using Genetic Programming with this representation was substantially better than with its predecessor (Interaction-Transformation) and ranked close to the state-of-the-art on a contemporary Symbolic Regression benchmark. On a closer look at these results, we observed that the performance could be further improved with an additional selective pressure for smaller expressions when the dataset contains just a few data points. The introduction of a penalization term applied to the fitness measure improved the results on these smaller datasets. One problem with this approach is that it introduces two additional hyperparameters: i) a criteria to when the penalization should be activated and, ii) the amount of penalization to the fitness function. In this paper, we extend Transformation-Interaction-Rational to support multi-objective optimization, specifically the NSGA-II algorithm, and apply that to the same benchmark. A detailed analysis of the results show that the use of multi-objective optimization benefits the overall performance on a subset of the benchmarks while keeping the results similar to the single-objective approach on the remainder of the datasets. Specifically to the small datasets, we observe a small (and statistically insignificant) improvement of the results suggesting that further strategies must be explored.

An Investigation on the Potential of KAN in Speech Enhancement

High-fidelity speech enhancement often requires sophisticated modeling to capture intricate, multiscale patterns. Standard activation functions, while introducing nonlinearity, lack the flexibility to fully address this complexity. Kolmogorov-Arnold Networks (KAN), an emerging methodology that employs learnable activation functions on graph edges, present a promising alternative. This work investigates two novel KAN variants based on rational and radial basis functions for speech enhancement. We integrate the rational variant into the 1D CNN blocks of Demucs and the GRU-Transformer blocks of MP-SENet, while the radial variant is adapted to the 2D CNN-based decoders of MP-SENet. Experiments on the VoiceBank-DEMAND dataset show that replacing standard activations with KAN-based activations improves speech quality across both the time-domain and time-frequency domain methods with minimal impact on model size and FLOP, underscoring KAN's potential to improve speech enhancement models.

Order Theory in the Context of Machine Learning: an application

The paper ``Tropical Geometry of Deep Neural Networks'' by L. Zhang et al. introduces an equivalence between integer-valued neural networks (IVNN) with activation $\text{ReLU}_{t}$ and tropical rational functions, which come with a map to polytopes. Here, IVNN refers to a network with integer weights but real biases, and $\text{ReLU}_{t}$ is defined as $\text{ReLU}_{t}(x)=\max(x,t)$ for $t\in\mathbb{R}\cup\{-\infty\}$. For every poset with $n$ points, there exists a corresponding order polytope, i.e., a convex polytope in the unit cube $[0,1]^n$ whose coordinates obey the inequalities of the poset. We study neural networks whose associated polytope is an order polytope. We then explain how posets with four points induce neural networks that can be interpreted as $2\times 2$ convolutional filters. These poset filters can be added to any neural network, not only IVNN. Similarly to maxout, poset convolutional filters update the weights of the neural network during backpropagation with more precision than average pooling, max pooling, or mixed pooling, without the need to train extra parameters. We report experiments that support our statements. We also prove that the assignment from a poset to an order polytope (and to certain tropical polynomials) is one to one, and we define the structure of algebra over the operad of posets on tropical polynomials.

Modeling Task Immersion based on Goal Activation Mechanism

Immersion in a task is a prerequisite for creativity. However, excessive arousal in a single task has drawbacks, such as overlooking events outside of the task. To examine such a negative aspect, this study constructs a computational model of arousal dynamics where the excessively increased arousal makes the task transition difficult. The model was developed using functions integrated into the cognitive architecture Adaptive Control of Thought-Rational (ACT-R). Under the framework, arousal is treated as a coefficient affecting the overall activation level in the model. In our simulations, we set up two conditions demanding low and high arousal, trying to replicate corresponding human experiments. In each simulation condition, two sets of ACT-R parameters were assumed from the different interpretations of the human experimental settings. The results showed consistency of behavior between humans and models both in the two different simulation settings. This result suggests the validity of our assumptions and has implications of controlling arousal in our daily life.

Retrieval-Enhanced Mutation Mastery: Augmenting Zero-Shot Prediction of Protein Language Model

Enzyme engineering enables the modification of wild-type proteins to meet industrial and research demands by enhancing catalytic activity, stability, binding affinities, and other properties. The emergence of deep learning methods for protein modeling has demonstrated superior results at lower costs compared to traditional approaches such as directed evolution and rational design. In mutation effect prediction, the key to pre-training deep learning models lies in accurately interpreting the complex relationships among protein sequence, structure, and function. This study introduces a retrieval-enhanced protein language model for comprehensive analysis of native properties from sequence and local structural interactions, as well as evolutionary properties from retrieved homologous sequences. The state-of-the-art performance of the proposed ProtREM is validated on over 2 million mutants across 217 assays from an open benchmark (ProteinGym). We also conducted post-hoc analyses of the model's ability to improve the stability and binding affinity of a VHH antibody. Additionally, we designed 10 new mutants on a DNA polymerase and conducted wet-lab experiments to evaluate their enhanced activity at higher temperatures. Both in silico and experimental evaluations confirmed that our method provides reliable predictions of mutation effects, offering an auxiliary tool for biologists aiming to evolve existing enzymes. The implementation is publicly available at https://github.com/tyang816/ProtREM.

Mitigating the Stability-Plasticity Dilemma in Adaptive Train Scheduling with Curriculum-Driven Continual DQN Expansion

A continual learning agent builds on previous experiences to develop increasingly complex behaviors by adapting to non-stationary and dynamic environments while preserving previously acquired knowledge. However, scaling these systems presents significant challenges, particularly in balancing the preservation of previous policies with the adaptation of new ones to current environments. This balance, known as the stability-plasticity dilemma, is especially pronounced in complex multi-agent domains such as the train scheduling problem, where environmental and agent behaviors are constantly changing, and the search space is vast. In this work, we propose addressing these challenges in the train scheduling problem using curriculum learning. We design a curriculum with adjacent skills that build on each other to improve generalization performance. Introducing a curriculum with distinct tasks introduces non-stationarity, which we address by proposing a new algorithm: Continual Deep Q-Network (DQN) Expansion (CDE). Our approach dynamically generates and adjusts Q-function subspaces to handle environmental changes and task requirements. CDE mitigates catastrophic forgetting through EWC while ensuring high plasticity using adaptive rational activation functions. Experimental results demonstrate significant improvements in learning efficiency and adaptability compared to RL baselines and other adapted methods for continual learning, highlighting the potential of our method in managing the stability-plasticity dilemma in the adaptive train scheduling setting.