Parallel Solution of Nonlinear Projection Equations in a Multitask Learning Framework.

Nonlinear projection equations (NPEs) provide a unified framework for addressing various constrained nonlinear optimization and engineering problems. However, when it comes to solving multiple NPEs, traditional numerical integration methods are not efficient enough. This is because traditional methods solve each NPE iteratively and independently. In this article, we propose a novel approach based on multitask learning (MTL) for solving multiple NPEs. The solution procedure is outlined as follows. First, we model each NPE as a system of ordinary differential equations (ODEs) using neurodynamic optimization. Second, for each ODE system, we use a physics-informed neural network (PINN) as the solution. Third, we use a multibranch MTL framework, where each branch corresponds to a PINN model. This allows us to solve multiple NPEs in parallel by training a single neural network model. Experimental results show that our approach has superior computational performance, especially when the number of NPEs to be solved is large.

Enhancing neurodynamic approach with physics-informed neural networks for solving non-smooth convex optimization problems.

This paper proposes a deep learning approach for solving non-smooth convex optimization problems (NCOPs), which have broad applications in computer science, engineering, and physics. Our approach combines neurodynamic optimization with physics-informed neural networks (PINNs) to provide an efficient and accurate solution. We first use neurodynamic optimization to formulate an initial value problem (IVP) that involves a system of ordinary differential equations for the NCOP. We then introduce a modified PINN as an approximate state solution to the IVP. Finally, we develop a dedicated algorithm to train the model to solve the IVP and minimize the NCOP objective simultaneously. Unlike existing numerical integration methods, a key advantage of our approach is that it does not require the computation of a series of intermediate states to produce a prediction of the NCOP. Our experimental results show that this computational feature results in fewer iterations being required to produce more accurate prediction solutions. Furthermore, our approach is effective in finding feasible solutions that satisfy the NCOP constraint.

MG-CNN: A deep CNN to predict saddle points of matrix games.

Finding the saddle point of a matrix game is a classical problem that arises in various fields, e.g., economics, computer science, and engineering. The standard problem-solving methods consist of formulating the problem as a linear program (LP). However, this approach seems less efficient when many instances need to be solved. In this paper, we propose a Convolutional Neural Network based approach, which is able to predict both the strategy profile (x,y) and the optimal value v of the game. We call this approach Matrix Game-Conventional Neural Network or MG-CNN for short. Thanks to a global pooling technique, MG-CNN can solve matrix games with different shapes. We propose a specialized algorithm to train MG-CNN, which includes both data generation and model training. Our numerical experiments show that MG-CNN outperforms standard LP solvers in terms of computational CPU time and provides a high-quality prediction.

A dynamical neural network approach for solving stochastic two-player zero-sum games.

This paper aims at solving a stochastic two-player zero-sum Nash game problem studied in Singh and Lisser (2019). The main contribution of our paper is that we model this game problem as a dynamical neural network (DNN for short). In this paper, we show that the saddle point of this game problem is the equilibrium point of the DNN model, and we study the globally asymptotically stable of the DNN model. In our numerical experiments, we present the time-continuous feature of the DNN model and compare it with the state-of-the-art convex solvers, i.e., Splitting conic solver (SCS for short) and Cvxopt. Our numerical results show that our DNN method has two advantages in dealing with this game problem. Firstly, the DNN model can converge to a better optimal point. Secondly, the DNN method can solve all problems, even when the problem size is large.

Ordering Protein Contact Matrices.

Numerous biophysical approaches provide information about residues spatial proximity in proteins. However, correct assignment of the protein fold from this proximity information is not straightforward if the spatially close protein residues are not assigned to residues in the primary sequence. Here, we propose an algorithm to assign such residue numbers by ordering the columns and lines of the raw protein contact matrix directly obtained from proximity information between unassigned amino acids. The ordering problem is formatted as the search of a trail within a graph connecting protein residues through the nonzero contact values. The algorithm performs in two steps: (i) finding the longest trail of the graph using an original dynamic programming algorithm, (ii) clustering the individual ordered matrices using a self-organizing map (SOM) approach. The combination of the dynamic programming and self-organizing map approaches constitutes a quite innovative point of the present work. The algorithm was validated on a set of about 900 proteins, representative of the sizes and proportions of secondary structures observed in the Protein Data Bank. The algorithm was revealed to be efficient for noise levels up to 40%, obtaining average gaps of about 20% at maximum between ordered and initial matrices. The proposed approach paves the ways toward a method of fold prediction from noisy proximity information, as TM scores larger than 0.5 have been obtained for ten randomly chosen proteins, in the case of a noise level of 10%. The methods has been also validated on two experimental cases, on which it performed satisfactorily.