Daily Papers

Large Language Models (LLM) are prone to returning false information. It constitutes one of major challenges in the AI field. In our work, we explore paradigm introduced by Inference-Time-Intervention (ITI). In first stage, it identifies attention heads, which contain the highest amount of desired type of knowledge (e.g., truthful). Afterwards, during inference, LLM activations are shifted for chosen subset of attention heads. We further improved the ITI framework by introducing a nonlinear probing and multi-token intervention - Non-Linear ITI (NL-ITI). NL-ITI is tested on diverse multiple-choice benchmarks, including TruthfulQA, on which we report around 14% MC1 metric improvement with respect to the baseline ITI results. NL-ITI achieves also encouraging results on other testsets - on Business Ethics subdomain of MMLU, around 18% MC1 improvement over baseline LLaMA2-7B. Additionally, NL-ITI performs better while being less invasive in the behavior of LLM at the same time (as measured by Kullback-Leibler divergence).

Multiplications are responsible for most of the computational cost involved in neural network training and inference. Recent research has thus looked for ways to reduce the cost associated with them. Inspired by Mogami (2020), we replace multiplication with a cheap piecewise affine approximation that is achieved by adding the bit representation of the floating point numbers together as integers. We show that transformers can be trained with the resulting modified matrix multiplications on both vision and language tasks with little to no performance impact, and without changes to the training hyperparameters. We further replace all non-linearities in the networks making them fully and jointly piecewise affine in both inputs and weights. Finally, we show that we can eliminate all multiplications in the entire training process, including operations in the forward pass, backward pass and optimizer update, demonstrating the first successful training of modern neural network architectures in a fully multiplication-free fashion.

We introduce a novel machine learning model for credit risk by combining tree-boosting with a latent spatio-temporal Gaussian process model accounting for frailty correlation. This allows for modeling non-linearities and interactions among predictor variables in a flexible data-driven manner and for accounting for spatio-temporal variation that is not explained by observable predictor variables. We also show how estimation and prediction can be done in a computationally efficient manner. In an application to a large U.S. mortgage credit risk data set, we find that both predictive default probabilities for individual loans and predictive loan portfolio loss distributions obtained with our novel approach are more accurate compared to conventional independent linear hazard models and also linear spatio-temporal models. Using interpretability tools for machine learning models, we find that the likely reasons for this outperformance are strong interaction and non-linear effects in the predictor variables and the presence of large spatio-temporal frailty effects.

Stochastic neurons and hard non-linearities can be useful for a number of reasons in deep learning models, but in many cases they pose a challenging problem: how to estimate the gradient of a loss function with respect to the input of such stochastic or non-smooth neurons? I.e., can we "back-propagate" through these stochastic neurons? We examine this question, existing approaches, and compare four families of solutions, applicable in different settings. One of them is the minimum variance unbiased gradient estimator for stochatic binary neurons (a special case of the REINFORCE algorithm). A second approach, introduced here, decomposes the operation of a binary stochastic neuron into a stochastic binary part and a smooth differentiable part, which approximates the expected effect of the pure stochatic binary neuron to first order. A third approach involves the injection of additive or multiplicative noise in a computational graph that is otherwise differentiable. A fourth approach heuristically copies the gradient with respect to the stochastic output directly as an estimator of the gradient with respect to the sigmoid argument (we call this the straight-through estimator). To explore a context where these estimators are useful, we consider a small-scale version of {\em conditional computation}, where sparse stochastic units form a distributed representation of gaters that can turn off in combinatorially many ways large chunks of the computation performed in the rest of the neural network. In this case, it is important that the gating units produce an actual 0 most of the time. The resulting sparsity can be potentially be exploited to greatly reduce the computational cost of large deep networks for which conditional computation would be useful.

Contours or closed planar curves are common in many domains. For example, they appear as object boundaries in computer vision, isolines in meteorology, and the orbits of rotating machinery. In many cases when learning from contour data, planar rotations of the input will result in correspondingly rotated outputs. It is therefore desirable that deep learning models be rotationally equivariant. In addition, contours are typically represented as an ordered sequence of edge points, where the choice of starting point is arbitrary. It is therefore also desirable for deep learning methods to be equivariant under cyclic shifts. We present RotaTouille, a deep learning framework for learning from contour data that achieves both rotation and cyclic shift equivariance through complex-valued circular convolution. We further introduce and characterize equivariant non-linearities, coarsening layers, and global pooling layers to obtain invariant representations for downstream tasks. Finally, we demonstrate the effectiveness of RotaTouille through experiments in shape classification, reconstruction, and contour regression.

Convolutional networks are large linear systems divided into layers and connected by non-linear units. These units are the "articulations" that allow the network to adapt to the input. To understand how a network manages to solve a problem we must look at the articulated decisions in entirety. If we could capture the actions of non-linear units for a particular input, we would be able to replay the whole system back and forth as if it was always linear. It would also reveal the actions of non-linearities because the resulting linear system, a Linear Interpreter, depends on the input image. We introduce a hooking layer, called a LinearScope, which allows us to run the network and the linear interpreter in parallel. Its implementation is simple, flexible and efficient. From here we can make many curious inquiries: how do these linear systems look like? When the rows and columns of the transformation matrix are images, how do they look like? What type of basis do these linear transformations rely on? The answers depend on the problems presented, through which we take a tour to some popular architectures used for classification, super-resolution (SR) and image-to-image translation (I2I). For classification we observe that popular networks use a pixel-wise vote per class strategy and heavily rely on bias parameters. For SR and I2I we find that CNNs use wavelet-type basis similar to the human visual system. For I2I we reveal copy-move and template-creation strategies to generate outputs.

The recent advancements in deep learning have brought about significant changes in various aspects of people's lives. Meanwhile, these rapid developments have raised concerns about the legitimacy of the training process of deep neural networks. To protect the intellectual properties of AI developers, directly examining the training process by accessing the model parameters and training data is often prohibited for verifiers. In response to this challenge, we present zero-knowledge deep learning (zkDL), an efficient zero-knowledge proof for deep learning training. To address the long-standing challenge of verifiable computations of non-linearities in deep learning training, we introduce zkReLU, a specialized proof for the ReLU activation and its backpropagation. zkReLU turns the disadvantage of non-arithmetic relations into an advantage, leading to the creation of FAC4DNN, our specialized arithmetic circuit design for modelling neural networks. This design aggregates the proofs over different layers and training steps, without being constrained by their sequential order in the training process. With our new CUDA implementation that achieves full compatibility with the tensor structures and the aggregated proof design, zkDL enables the generation of complete and sound proofs in less than a second per batch update for an 8-layer neural network with 10M parameters and a batch size of 64, while provably ensuring the privacy of data and model parameters. To our best knowledge, we are not aware of any existing work on zero-knowledge proof of deep learning training that is scalable to million-size networks.

A proper initialization of the weights in a neural network is critical to its convergence. Current insights into weight initialization come primarily from linear activation functions. In this paper, I develop a theory for weight initializations with non-linear activations. First, I derive a general weight initialization strategy for any neural network using activation functions differentiable at 0. Next, I derive the weight initialization strategy for the Rectified Linear Unit (RELU), and provide theoretical insights into why the Xavier initialization is a poor choice with RELU activations. My analysis provides a clear demonstration of the role of non-linearities in determining the proper weight initializations.

Deep convolutional neural networks have led to breakthrough results in numerous practical machine learning tasks such as classification of images in the ImageNet data set, control-policy-learning to play Atari games or the board game Go, and image captioning. Many of these applications first perform feature extraction and then feed the results thereof into a trainable classifier. The mathematical analysis of deep convolutional neural networks for feature extraction was initiated by Mallat, 2012. Specifically, Mallat considered so-called scattering networks based on a wavelet transform followed by the modulus non-linearity in each network layer, and proved translation invariance (asymptotically in the wavelet scale parameter) and deformation stability of the corresponding feature extractor. This paper complements Mallat's results by developing a theory that encompasses general convolutional transforms, or in more technical parlance, general semi-discrete frames (including Weyl-Heisenberg filters, curvelets, shearlets, ridgelets, wavelets, and learned filters), general Lipschitz-continuous non-linearities (e.g., rectified linear units, shifted logistic sigmoids, hyperbolic tangents, and modulus functions), and general Lipschitz-continuous pooling operators emulating, e.g., sub-sampling and averaging. In addition, all of these elements can be different in different network layers. For the resulting feature extractor we prove a translation invariance result of vertical nature in the sense of the features becoming progressively more translation-invariant with increasing network depth, and we establish deformation sensitivity bounds that apply to signal classes such as, e.g., band-limited functions, cartoon functions, and Lipschitz functions.

In this work we introduce Steerable Transformers, an extension of the Vision Transformer mechanism that maintains equivariance to the special Euclidean group SE(d). We propose an equivariant attention mechanism that operates on features extracted by steerable convolutions. Operating in Fourier space, our network utilizes Fourier space non-linearities. Our experiments in both two and three dimensions show that adding a steerable transformer encoder layer to a steerable convolution network enhances performance.

In this paper we describe a new mobile architecture, MobileNetV2, that improves the state of the art performance of mobile models on multiple tasks and benchmarks as well as across a spectrum of different model sizes. We also describe efficient ways of applying these mobile models to object detection in a novel framework we call SSDLite. Additionally, we demonstrate how to build mobile semantic segmentation models through a reduced form of DeepLabv3 which we call Mobile DeepLabv3. The MobileNetV2 architecture is based on an inverted residual structure where the input and output of the residual block are thin bottleneck layers opposite to traditional residual models which use expanded representations in the input an MobileNetV2 uses lightweight depthwise convolutions to filter features in the intermediate expansion layer. Additionally, we find that it is important to remove non-linearities in the narrow layers in order to maintain representational power. We demonstrate that this improves performance and provide an intuition that led to this design. Finally, our approach allows decoupling of the input/output domains from the expressiveness of the transformation, which provides a convenient framework for further analysis. We measure our performance on Imagenet classification, COCO object detection, VOC image segmentation. We evaluate the trade-offs between accuracy, and number of operations measured by multiply-adds (MAdd), as well as the number of parameters

Learning-based methods, particularly Reinforcement Learning (RL), hold great promise for streamlining deployment, enhancing performance, and achieving generalization in the control of autonomous multirotor aerial vehicles. Deep RL has been able to control complex systems with impressive fidelity and agility in simulation but the simulation-to-reality transfer often brings a hard-to-bridge reality gap. Moreover, RL is commonly plagued by prohibitively long training times. In this work, we propose a novel asymmetric actor-critic-based architecture coupled with a highly reliable RL-based training paradigm for end-to-end quadrotor control. We show how curriculum learning and a highly optimized simulator enhance sample complexity and lead to fast training times. To precisely discuss the challenges related to low-level/end-to-end multirotor control, we also introduce a taxonomy that classifies the existing levels of control abstractions as well as non-linearities and domain parameters. Our framework enables Simulation-to-Reality (Sim2Real) transfer for direct RPM control after only 18 seconds of training on a consumer-grade laptop as well as its deployment on microcontrollers to control a multirotor under real-time guarantees. Finally, our solution exhibits competitive performance in trajectory tracking, as demonstrated through various experimental comparisons with existing state-of-the-art control solutions using a real Crazyflie nano quadrotor. We open source the code including a very fast multirotor dynamics simulator that can simulate about 5 months of flight per second on a laptop GPU. The fast training times and deployment to a cheap, off-the-shelf quadrotor lower the barriers to entry and help democratize the research and development of these systems.

Many applications of cross-modal music retrieval are related to connecting sheet music images to audio recordings. A typical and recent approach to this is to learn, via deep neural networks, a joint embedding space that correlates short fixed-size snippets of audio and sheet music by means of an appropriate similarity structure. However, two challenges that arise out of this strategy are the requirement of strongly aligned data to train the networks, and the inherent discrepancies of musical content between audio and sheet music snippets caused by local and global tempo differences. In this paper, we address these two shortcomings by designing a cross-modal recurrent network that learns joint embeddings that can summarize longer passages of corresponding audio and sheet music. The benefits of our method are that it only requires weakly aligned audio-sheet music pairs, as well as that the recurrent network handles the non-linearities caused by tempo variations between audio and sheet music. We conduct a number of experiments on synthetic and real piano data and scores, showing that our proposed recurrent method leads to more accurate retrieval in all possible configurations.

Although CNNs are believed to be invariant to translations, recent works have shown this is not the case, due to aliasing effects that stem from downsampling layers. The existing architectural solutions to prevent aliasing are partial since they do not solve these effects, that originate in non-linearities. We propose an extended anti-aliasing method that tackles both downsampling and non-linear layers, thus creating truly alias-free, shift-invariant CNNs. We show that the presented model is invariant to integer as well as fractional (i.e., sub-pixel) translations, thus outperforming other shift-invariant methods in terms of robustness to adversarial translations.

Neural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, f:XtoY. Is it possible to identify a pair of non-standard vector spaces for which a conventionally nonlinear function is, in fact, linear? This paper introduces a method that makes such vector spaces explicit by construction. We find that if we sandwich a linear operator A between two invertible neural networks, f(x)=g_y^{-1}(A g_x(x)), then the corresponding vector spaces X and Y are induced by newly defined addition and scaling actions derived from g_x and g_y. We term this kind of architecture a Linearizer. This framework makes the entire arsenal of linear algebra, including SVD, pseudo-inverse, orthogonal projection and more, applicable to nonlinear mappings. Furthermore, we show that the composition of two Linearizers that share a neural network is also a Linearizer. We leverage this property and demonstrate that training diffusion models using our architecture makes the hundreds of sampling steps collapse into a single step. We further utilize our framework to enforce idempotency (i.e. f(f(x))=f(x)) on networks leading to a globally projective generative model and to demonstrate modular style transfer.

Recent studies of gradient descent with large step sizes have shown that there is often a regime with an initial increase in the largest eigenvalue of the loss Hessian (progressive sharpening), followed by a stabilization of the eigenvalue near the maximum value which allows convergence (edge of stability). These phenomena are intrinsically non-linear and do not happen for models in the constant Neural Tangent Kernel (NTK) regime, for which the predictive function is approximately linear in the parameters. As such, we consider the next simplest class of predictive models, namely those that are quadratic in the parameters, which we call second-order regression models. For quadratic objectives in two dimensions, we prove that this second-order regression model exhibits progressive sharpening of the NTK eigenvalue towards a value that differs slightly from the edge of stability, which we explicitly compute. In higher dimensions, the model generically shows similar behavior, even without the specific structure of a neural network, suggesting that progressive sharpening and edge-of-stability behavior aren't unique features of neural networks, and could be a more general property of discrete learning algorithms in high-dimensional non-linear models.

The majority of quantization methods have been proposed to reduce the model size of Vision Transformers, yet most of them have overlooked the quantization of non-linear operations. Only a few works have addressed quantization for non-linear operations, but they applied a single quantization method across all non-linear operations. We believe that this can be further improved by employing a different quantization method for each non-linear operation. Therefore, to assign the most error-minimizing quantization method from the known methods to each non-linear layer, we propose a mixed non-linear quantization that considers layer-wise quantization sensitivity measured by SQNR difference metric. The results show that our method outperforms I-BERT, FQ-ViT, and I-ViT in both 8-bit and 6-bit settings for ViT, DeiT, and Swin models by an average of 0.6%p and 19.6%p, respectively. Our method outperforms I-BERT and I-ViT by 0.6%p and 20.8%p, respectively, when training time is limited. We plan to release our code at https://gitlab.com/ones-ai/mixed-non-linear-quantization.

The analysis of parametric and non-parametric uncertainties of very large dynamical systems requires the construction of a stochastic model of said system. Linear approaches relying on random matrix theory and principal componant analysis can be used when systems undergo low-frequency vibrations. In the case of fast dynamics and wave propagation, we investigate a random generator of boundary conditions for fast submodels by using machine learning. We show that the use of non-linear techniques in machine learning and data-driven methods is highly relevant. Physics-informed neural networks is a possible choice for a data-driven method to replace linear modal analysis. An architecture that support a random component is necessary for the construction of the stochastic model of the physical system for non-parametric uncertainties, since the goal is to learn the underlying probabilistic distribution of uncertainty in the data. Generative Adversarial Networks (GANs) are suited for such applications, where the Wasserstein-GAN with gradient penalty variant offers improved convergence results for our problem. The objective of our approach is to train a GAN on data from a finite element method code (Fenics) so as to extract stochastic boundary conditions for faster finite element predictions on a submodel. The submodel and the training data have both the same geometrical support. It is a zone of interest for uncertainty quantification and relevant to engineering purposes. In the exploitation phase, the framework can be viewed as a randomized and parametrized simulation generator on the submodel, which can be used as a Monte Carlo estimator.

We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.

Existing works have extensively studied adversarial examples, which are minimal perturbations that can mislead the output of deep neural networks (DNNs) while remaining imperceptible to humans. However, in this work, we reveal the existence of a harmless perturbation space, in which perturbations drawn from this space, regardless of their magnitudes, leave the network output unchanged when applied to inputs. Essentially, the harmless perturbation space emerges from the usage of non-injective functions (linear or non-linear layers) within DNNs, enabling multiple distinct inputs to be mapped to the same output. For linear layers with input dimensions exceeding output dimensions, any linear combination of the orthogonal bases of the nullspace of the parameter consistently yields no change in their output. For non-linear layers, the harmless perturbation space may expand, depending on the properties of the layers and input samples. Inspired by this property of DNNs, we solve for a family of general perturbation spaces that are redundant for the DNN's decision, and can be used to hide sensitive data and serve as a means of model identification. Our work highlights the distinctive robustness of DNNs (i.e., consistency under large magnitude perturbations) in contrast to adversarial examples (vulnerability for small imperceptible noises).

Implicit Neural Representation (INR), leveraging a neural network to transform coordinate input into corresponding attributes, has recently driven significant advances in several vision-related domains. However, the performance of INR is heavily influenced by the choice of the nonlinear activation function used in its multilayer perceptron (MLP) architecture. Multiple nonlinearities have been investigated; yet, current INRs face limitations in capturing high-frequency components, diverse signal types, and handling inverse problems. We have identified that these problems can be greatly alleviated by introducing a paradigm shift in INRs. We find that an architecture with learnable activations in initial layers can represent fine details in the underlying signals. Specifically, we propose SL^{2}A-INR, a hybrid network for INR with a single-layer learnable activation function, prompting the effectiveness of traditional ReLU-based MLPs. Our method performs superior across diverse tasks, including image representation, 3D shape reconstructions, inpainting, single image super-resolution, CT reconstruction, and novel view synthesis. Through comprehensive experiments, SL^{2}A-INR sets new benchmarks in accuracy, quality, and convergence rates for INR.

Dimensionality reduction techniques are fundamental for analyzing and visualizing high-dimensional data. With established methods like t-SNE and PCA presenting a trade-off between representational power and interpretability. This paper introduces a novel approach that bridges this gap by combining the interpretability of linear methods with the expressiveness of non-linear transformations. The proposed algorithm constructs a non-linear mapping between high-dimensional and low-dimensional spaces through a combination of linear transformations, each weighted by Gaussian functions. This architecture enables complex non-linear transformations while preserving the interpretability advantages of linear methods, as each transformation can be analyzed independently. The resulting model provides both powerful dimensionality reduction and transparent insights into the transformed space. Techniques for interpreting the learned transformations are presented, including methods for identifying suppressed dimensions and how space is expanded and contracted. These tools enable practitioners to understand how the algorithm preserves and modifies geometric relationships during dimensionality reduction. To ensure the practical utility of this algorithm, the creation of user-friendly software packages is emphasized, facilitating its adoption in both academia and industry.

Non-linear activation functions are crucial in Convolutional Neural Networks. However, until now they have not been well described in the frequency domain. In this work, we study the spectral behavior of ReLU, a popular activation function. We use the ReLU's Taylor expansion to derive its frequency domain behavior. We demonstrate that ReLU introduces higher frequency oscillations in the signal and a constant DC component. Furthermore, we investigate the importance of this DC component, where we demonstrate that it helps the model extract meaningful features related to the input frequency content. We accompany our theoretical derivations with experiments and real-world examples. First, we numerically validate our frequency response model. Then we observe ReLU's spectral behavior on two example models and a real-world one. Finally, we experimentally investigate the role of the DC component introduced by ReLU in the CNN's representations. Our results indicate that the DC helps to converge to a weight configuration that is close to the initial random weights.

In this study, we present a technique that spans multi-scale views (global scale -- meaning brain network-level and local scale -- examining each individual ROI that constitutes the network) applied to resting-state fMRI volumes. Deep learning based classification is utilized in understanding neurodegeneration. The novelty of the proposed approach lies in utilizing two extreme scales of analysis. One branch considers the entire network within graph-analysis framework. Concurrently, the second branch scrutinizes each ROI within a network independently, focusing on evolution of dynamics. For each subject, graph-based approach employs partial correlation to profile the subject in a single graph where each ROI is a node, providing insights into differences in levels of participation. In contrast, non-linear analysis employs recurrence plots to profile a subject as a multichannel 2D image, revealing distinctions in underlying dynamics. The proposed approach is employed for classification of a cohort of 50 healthy control (HC) and 50 Mild Cognitive Impairment (MCI), sourced from ADNI dataset. Results point to: (1) reduced activity in ROIs such as PCC in MCI (2) greater activity in occipital in MCI, which is not seen in HC (3) when analysed for dynamics, all ROIs in MCI show greater predictability in time-series.

Neural networks have proven to be a highly effective tool for solving complex problems in many areas of life. Recently, their importance and practical usability have further been reinforced with the advent of deep learning. One of the important conditions for the success of neural networks is the choice of an appropriate activation function introducing non-linearity into the model. Many types of these functions have been proposed in the literature in the past, but there is no single comprehensive source containing their exhaustive overview. The absence of this overview, even in our experience, leads to redundancy and the unintentional rediscovery of already existing activation functions. To bridge this gap, our paper presents an extensive survey involving 400 activation functions, which is several times larger in scale than previous surveys. Our comprehensive compilation also references these surveys; however, its main goal is to provide the most comprehensive overview and systematization of previously published activation functions with links to their original sources. The secondary aim is to update the current understanding of this family of functions.

We introduce Tangent Attention Fine-Tuning (TAFT), a method for fine-tuning linearized transformers obtained by computing a First-order Taylor Expansion around a pre-trained initialization. We show that the Jacobian-Vector Product resulting from linearization can be computed efficiently in a single forward pass, reducing training and inference cost to the same order of magnitude as its original non-linear counterpart, while using the same number of parameters. Furthermore, we show that, when applied to various downstream visual classification tasks, the resulting Tangent Transformer fine-tuned with TAFT can perform comparably with fine-tuning the original non-linear network. Since Tangent Transformers are linear with respect to the new set of weights, and the resulting fine-tuning loss is convex, we show that TAFT enjoys several advantages compared to non-linear fine-tuning when it comes to model composition, parallel training, machine unlearning, and differential privacy.

In Neural Architecture Search (NAS), reducing the cost of architecture evaluation remains one of the most crucial challenges. Among a plethora of efforts to bypass training of each candidate architecture to convergence for evaluation, the Neural Tangent Kernel (NTK) is emerging as a promising theoretical framework that can be utilized to estimate the performance of a neural architecture at initialization. In this work, we revisit several at-initialization metrics that can be derived from the NTK and reveal their key shortcomings. Then, through the empirical analysis of the time evolution of NTK, we deduce that modern neural architectures exhibit highly non-linear characteristics, making the NTK-based metrics incapable of reliably estimating the performance of an architecture without some amount of training. To take such non-linear characteristics into account, we introduce Label-Gradient Alignment (LGA), a novel NTK-based metric whose inherent formulation allows it to capture the large amount of non-linear advantage present in modern neural architectures. With minimal amount of training, LGA obtains a meaningful level of rank correlation with the post-training test accuracy of an architecture. Lastly, we demonstrate that LGA, complemented with few epochs of training, successfully guides existing search algorithms to achieve competitive search performances with significantly less search cost. The code is available at: https://github.com/nutellamok/DemystifyingNTK.

Initial value problems -- a system of ordinary differential equations and corresponding initial conditions -- can be used to describe many physical phenomena including those arise in classical mechanics. We have developed a novel approach to solve physics-based initial value problems using unsupervised machine learning. We propose a deep learning framework that models the dynamics of a variety of mechanical systems through neural networks. Our framework is flexible, allowing us to solve non-linear, coupled, and chaotic dynamical systems. We demonstrate the effectiveness of our approach on systems including a free particle, a particle in a gravitational field, a classical pendulum, and the H\'enon--Heiles system (a pair of coupled harmonic oscillators with a non-linear perturbation, used in celestial mechanics). Our results show that deep neural networks can successfully approximate solutions to these problems, producing trajectories which conserve physical properties such as energy and those with stationary action. We note that probabilistic activation functions, as defined in this paper, are required to learn any solutions of initial value problems in their strictest sense, and we introduce coupled neural networks to learn solutions of coupled systems.

Neural networks have shown tremendous growth in recent years to solve numerous problems. Various types of neural networks have been introduced to deal with different types of problems. However, the main goal of any neural network is to transform the non-linearly separable input data into more linearly separable abstract features using a hierarchy of layers. These layers are combinations of linear and nonlinear functions. The most popular and common non-linearity layers are activation functions (AFs), such as Logistic Sigmoid, Tanh, ReLU, ELU, Swish and Mish. In this paper, a comprehensive overview and survey is presented for AFs in neural networks for deep learning. Different classes of AFs such as Logistic Sigmoid and Tanh based, ReLU based, ELU based, and Learning based are covered. Several characteristics of AFs such as output range, monotonicity, and smoothness are also pointed out. A performance comparison is also performed among 18 state-of-the-art AFs with different networks on different types of data. The insights of AFs are presented to benefit the researchers for doing further research and practitioners to select among different choices. The code used for experimental comparison is released at: https://github.com/shivram1987/ActivationFunctions.

Lara is a key-value algebra that aims at unifying linear and relational algebra with three types of operation abstraction. The study of Lara's expressive ability reports that it can represent relational algebra and most linear algebra operations. However, several essential computations, such as matrix inversion and determinant, cannot be expressed in Lara. Lara cannot represent global and iterative computation, either. This article proposes IterLara, extending Lara with iterative operators, to provide an algebraic model that unifies operations in general-purpose computing, like big data, AI, scientific computing, and database. We study the expressive ability of Lara and IterLara and prove that IterLara with aggregation functions can represent matrix inversion, determinant. Besides, we demonstrate that IterLara with no limitation of function utility is Turing complete. We also propose the Operation Count (OP) as a metric of computation amount for IterLara and ensure that the OP metric is in accordance with the existing computation metrics.

We propose the APTx Neuron, a novel, unified neural computation unit that integrates non-linear activation and linear transformation into a single trainable expression. The APTx Neuron is derived from the APTx activation function, thereby eliminating the need for separate activation layers and making the architecture both computationally efficient and elegant. The proposed neuron follows the functional form y = sum_{i=1}^{n} ((alpha_i + tanh(beta_i x_i)) cdot gamma_i x_i) + delta, where all parameters alpha_i, beta_i, gamma_i, and delta are trainable. We validate our APTx Neuron-based architecture on the MNIST dataset, achieving up to 96.69% test accuracy within 11 epochs using approximately 332K trainable parameters. The results highlight the superior expressiveness and computational efficiency of the APTx Neuron compared to traditional neurons, pointing toward a new paradigm in unified neuron design and the architectures built upon it.

The LayerNorm (LN) layer in GPT-style transformer models has long been a hindrance to mechanistic interpretability. LN is a crucial component required to stabilize the training of large language models, and LN or the similar RMSNorm have been used in practically all large language models based on the transformer architecture. The non-linear nature of the LN layers is a hindrance for mechanistic interpretability as it hinders interpretation of the residual stream, and makes it difficult to decompose the model into circuits. Some research have gone so far as to name "reasons interpretability researchers hate layer norm". In this paper we show that it is possible to remove the LN layers from a pre-trained GPT2-small model by fine-tuning on a fraction (500M tokens) of the training data. We demonstrate that this LN-free model achieves similar performance to the original model on the OpenWebText and ThePile datasets (-0.05 cross-entropy loss), and the Hellaswag benchmark (-0.5% accuracy). We provide the fine-tuning procedure and a Hugging Face repository with the fine-tuned GPT2-small models. Our work not only provides a simplified model for mechanistic interpretability research, but also provides evidence that the LN layers, at inference time, do not play a crucial role in transformer models.