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1. Multi-grade Deep Learning

   https://doi.org/10.1007/s42967-024-00474-y  

   Xu, Yuesheng (March 2025, Communications on Applied Mathematics and Computation)

   Abstract Deep learning requires solving a nonconvex optimization problem of a large size to learn a deep neural network (DNN). The current deep learning model is of asingle-grade, that is, it trains a DNN end-to-end, by solving a single nonconvex optimization problem. When the layer number of the neural network is large, it is computationally challenging to carry out such a task efficiently. The complexity of the task comes from learning all weight matrices and bias vectors from one single nonconvex optimization problem of a large size. Inspired by the human education process which arranges learning in grades, we propose a multi-grade learning model: instead of solving one single optimization problem of a large size, we successively solve a number of optimization problems of small sizes, which are organized in grades, to learn a shallow neural network (a network having a few hidden layers) for each grade. Specifically, the current grade is to learn the leftover from the previous grade. In each of the grades, we learn a shallow neural network stacked on the top of the neural network, learned in the previous grades, whose parameters remain unchanged in training of the current and future grades. By dividing the task of learning a DDN into learning several shallow neural networks, one can alleviate the severity of the nonconvexity of the original optimization problem of a large size. When all grades of the learning are completed, the final neural network learned is astair-shapeneural network, which is thesuperpositionof networks learned from all grades. Such a model enables us to learn a DDN much more effectively and efficiently. Moreover, multi-grade learning naturally leads to adaptive learning. We prove that in the context of function approximation if the neural network generated by a new grade is nontrivial, the optimal error of a new grade is strictly reduced from the optimal error of the previous grade. Furthermore, we provide numerical examples which confirm that the proposed multi-grade model outperforms significantly the standard single-grade model and is much more robust to noise than the single-grade model. They include three proof-of-concept examples, classification on two benchmark data sets MNIST and Fashion MNIST with two noise rates, which is to find classifiers, functions of 784 dimensions, and as well as numerical solutions of the one-dimensional Helmholtz equation. 

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2. Guaranteed Quantization Error Computation for Neural Network Model Compression

   https://doi.org/10.1109/ICIT58465.2023.10143141  

   Cooke, Wesley; Mo, Zihao; Xiang, Weiming (April 2023, 2023 IEEE International Conference on Industrial Technology (ICIT))

   Neural network model compression techniques can address the computation issue of deep neural networks on embedded devices in industrial systems. The guaranteed output error computation problem for neural network compression with quantization is addressed in this paper. A merged neural network is built from a feedforward neural network and its quantized version to produce the exact output difference between two neural networks. Then, optimization-based methods and reachability analysis methods are applied to the merged neural network to compute the guaranteed quantization error. Finally, a numerical example is proposed to validate the applicability and effectiveness of the proposed approach. 

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3. Banach Space Representer Theorems for Neural Networks and Ridge Splines

   Parhi, Rahul; Nowak, Robert D (February 2021, Journal of machine learning research)

   We develop a variational framework to understand the properties of the functions learned by neural networks fit to data. We propose and study a family of continuous-domain linear inverse problems with total variation-like regularization in the Radon domain subject to data fitting constraints. We derive a representer theorem showing that finite-width, singlehidden layer neural networks are solutions to these inverse problems. We draw on many techniques from variational spline theory and so we propose the notion of polynomial ridge splines, which correspond to single-hidden layer neural networks with truncated power functions as the activation function. The representer theorem is reminiscent of the classical reproducing kernel Hilbert space representer theorem, but we show that the neural network problem is posed over a non-Hilbertian Banach space. While the learning problems are posed in the continuous-domain, similar to kernel methods, the problems can be recast as finite-dimensional neural network training problems. These neural network training problems have regularizers which are related to the well-known weight decay and path-norm regularizers. Thus, our result gives insight into functional characteristics of trained neural networks and also into the design neural network regularizers. We also show that these regularizers promote neural network solutions with desirable generalization properties. Keywords: neural networks, splines, inverse problems, regularization, sparsity 

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4. Harnessing deep neural networks to solve inverse problems in quantum dynamics: machine-learned predictions of time-dependent optimal control fields

   https://doi.org/10.1039/D0CP03694C  

   Wang, Xian; Kumar, Anshuman; Shelton, Christian R.; Wong, Bryan M. (October 2020, Physical Chemistry Chemical Physics)

   Inverse problems continue to garner immense interest in the physical sciences, particularly in the context of controlling desired phenomena in non-equilibrium systems. In this work, we utilize a series of deep neural networks for predicting time-dependent optimal control fields, E ( t ), that enable desired electronic transitions in reduced-dimensional quantum dynamical systems. To solve this inverse problem, we investigated two independent machine learning approaches: (1) a feedforward neural network for predicting the frequency and amplitude content of the power spectrum in the frequency domain ( i.e. , the Fourier transform of E ( t )), and (2) a cross-correlation neural network approach for directly predicting E ( t ) in the time domain. Both of these machine learning methods give complementary approaches for probing the underlying quantum dynamics and also exhibit impressive performance in accurately predicting both the frequency and strength of the optimal control field. We provide detailed architectures and hyperparameters for these deep neural networks as well as performance metrics for each of our machine-learned models. From these results, we show that machine learning, particularly deep neural networks, can be employed as cost-effective statistical approaches for designing electromagnetic fields to enable desired transitions in these quantum dynamical systems. 

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5. TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems

   https://doi.org/10.1137/22M1526708  

   Nguyen, Hai V; Bui-Thanh, Tan (February 2024, SIAM Journal on Scientific Computing)

   Deep Learning (DL), in particular deep neural networks (DNN), by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, and hence promoting better generalization. This paper develops a model-constrained deep learning approach and its variant TNet—a Tikhonov neural network—that are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solution with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions in any desired accuracy for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for 1D deconvolution, 5 for inverse 2D heat conductivity problem, 100 for inverse initial conditions for time-dependent 2D Burgers’ equation, and 50 for inverse initial conditions for 2D Navier-Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. This is possible owing to the model-constrained term, replications, and randomization. 

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