Neural Projection Filter: Learning Unknown Dynamics Driven by Noisy Observations.

In this article, we propose the novel neural stochastic differential equations (SDEs) driven by noisy sequential observations called neural projection filter (NPF) under the continuous state-space models (SSMs) framework. The contributions of this work are both theoretical and algorithmic. On the one hand, we investigate the approximation capacity of the NPF, i.e., the universal approximation theorem for NPF. More explicitly, under some natural assumptions, we prove that the solution of the SDE driven by the semimartingale can be well approximated by the solution of the NPF. In particular, the explicit estimation bound is given. On the other hand, as an important application of this result, we develop a novel data-driven filter based on NPF. Also, under certain condition, we prove the algorithm convergence; i.e., the dynamics of NPF converges to the target dynamics. At last, we systematically compare the NPF with the existing filters. We verify the convergence theorem in linear case and experimentally demonstrate that the NPF outperforms existing filters in nonlinear case with robustness and efficiency. Furthermore, NPF could handle high-dimensional systems in real-time manner, even for the 100 -D cubic sensor, while the state-of-the-art (SOTA) filter fails to do it.

Recurrent Neural Networks Are Universal Approximators With Stochastic Inputs.

In this article, we investigate the approximation ability of recurrent neural networks (RNNs) with stochastic inputs in state space model form. More explicitly, we prove that open dynamical systems with stochastic inputs can be well-approximated by a special class of RNNs under some natural assumptions, and the asymptotic approximation error has also been delicately analyzed as time goes to infinity. In addition, as an important application of this result, we construct an RNN-based filter and prove that it can well-approximate finite dimensional filters which include Kalman filter (KF) and Benes filter as special cases. The efficiency of RNN-based filter has also been verified by two numerical experiments compared with optimal KF.

Biomolecular Topology: Modelling and Analysis.

With the great advancement of experimental tools, a tremendous amount of biomolecular data has been generated and accumulated in various databases. The high dimensionality, structural complexity, the nonlinearity, and entanglements of biomolecular data, ranging from DNA knots, RNA secondary structures, protein folding configurations, chromosomes, DNA origami, molecular assembly, to others at the macromolecular level, pose a severe challenge in their analysis and characterization. In the past few decades, mathematical concepts, models, algorithms, and tools from algebraic topology, combinatorial topology, computational topology, and topological data analysis, have demonstrated great power and begun to play an essential role in tackling the biomolecular data challenge. In this work, we introduce biomolecular topology, which concerns the topological problems and models originated from the biomolecular systems. More specifically, the biomolecular topology encompasses topological structures, properties and relations that are emerged from biomolecular structures, dynamics, interactions, and functions. We discuss the various types of biomolecular topology from structures (of proteins, DNAs, and RNAs), protein folding, and protein assembly. A brief discussion of databanks (and databases), theoretical models, and computational algorithms, is presented. Further, we systematically review related topological models, including graphs, simplicial complexes, persistent homology, persistent Laplacians, de Rham-Hodge theory, Yau-Hausdorff distance, and the topology-based machine learning models.

New Genome Sequence Detection via Natural Vector Convex Hull Method.

It remains challenging how to find existing but undiscovered genome sequence mutations or predict potential genome sequence mutations based on real sequence data. Motivated by this, we develop approaches to detect new, undiscovered genome sequences. Because discovering new genome sequences through biological experiments is resource-intensive, we want to achieve the new genome sequence detection task mathematically. However, little literature tells us how to detect new, undiscovered genome sequence mutations mathematically. We form a new framework based on natural vector convex hull method that conducts alignment-free sequence analysis. Our newly developed two approaches, Random-permutation Algorithm with Penalty (RAP) and Random-permutation Algorithm with Penalty and COstrained Search (RAPCOS), use the geometry properties captured by natural vectors. In our experiment, we discover a mathematically new human immunodeficiency virus (HIV) genome sequence using some real HIV genome sequences. Significantly, the proposed methods are applicable to solve the new genome sequence detection challenge and have many good properties, such as robustness, rapid convergence, and fast computation.

Utilizing the codon adaptation index to evaluate the susceptibility to HIV-1 and SARS-CoV-2 related coronaviruses in possible target cells in humans.

Comprehensive identification of possible target cells for viruses is crucial for understanding the pathological mechanism of virosis. The susceptibility of cells to viruses depends on many factors. Besides the existence of receptors at the cell surface, effective expression of viral genes is also pivotal for viral infection. The regulation of viral gene expression is a multilevel process including transcription, translational initiation and translational elongation. At the translational elongation level, the translational efficiency of viral mRNAs mainly depends on the match between their codon composition and cellular translational machinery (usually referred to as codon adaptation). Thus, codon adaptation for viral ORFs in different cell types may be related to their susceptibility to viruses. In this study, we selected the codon adaptation index (CAI) which is a common codon adaptation-based indicator for assessing the translational efficiency at the translational elongation level to evaluate the susceptibility to two-pandemic viruses (HIV-1 and SARS-CoV-2) of different human cell types. Compared with previous studies that evaluated the infectivity of viruses based on codon adaptation, the main advantage of our study is that our analysis is refined to the cell-type level. At first, we verified the positive correlation between CAI and translational efficiency and strengthened the rationality of our research method. Then we calculated CAI for ORFs of two viruses in various human cell types. We found that compared to high-expression endogenous genes, the CAIs of viral ORFs are relatively low. This phenomenon implied that two kinds of viruses have not been well adapted to translational regulatory machinery in human cells. Also, we indicated that presumptive susceptibility to viruses according to CAI is usually consistent with the results of experimental research. However, there are still some exceptions. Finally, we found that two viruses have different effects on cellular translational mechanisms. HIV-1 decouples CAI and translational efficiency of endogenous genes in host cells and SARS-CoV-2 exhibits increased CAI for its ORFs in infected cells. Our results implied that at least in cases of HIV-1 and SARS-CoV-2, CAI can be regarded as an auxiliary index to assess cells' susceptibility to viruses but cannot be used as the only evidence to identify viral target cells.