Implicit Model Research Articles

Molecular dynamics simulation study of post‐transition state bifurcation: A case study on the ambimodal transition state of dipolar/Diels–Alder cycloaddition

AbstractThe potential energy surfaces for the reactions of 1,3‐butadiene with 2‐hydroxythioacrolein and 2‐aminoacrolein exhibit ambimodal transition states leading to both dipolar (4 + 3) and Diels–Alder (4 + 2) cycloaddition products, thereby demonstrating a post transition state bifurcation feature. We have investigated the bifurcation dynamics of these reactions using three molecular dynamics (MD) methods: quasi‐classical trajectory, classical MD, and ring‐polymer MD simulations. The trajectory calculations were performed with the semiempirical GFN2‐xTB method with the element‐specific parameters optimized to reproduce the density‐functional theory calculations. The effect of water solvation was examined using an implicit solvation model, revealing significant differences in bifurcation dynamic between gas‐phase and solution‐phase reactions. Nuclear quantum effects were found to play a crucial role in the proton‐transfer process from the (4 + 3) intermediate to the (4 + 3) product in the case of the 2‐aminoacrolein reaction.

Exploring Structural Insights of Aβ42 and α-Synuclein Monomers and Heterodimer: A Comparative Study Using Implicit and Explicit Solvent Simulations.

Protein misfolding, aggregation, and fibril formation play a central role in the development of severe neurological disorders, including Alzheimer's and Parkinson's diseases. The structural stability of mature fibrils in these diseases is of great importance, as organisms struggle to effectively eliminate amyloid plaques. To address this issue, it is crucial to investigate the early stages of fibril formation when monomers aggregate into small, toxic, and soluble oligomers. However, these structures are inherently disordered, making them challenging to study through experimental approaches. Recently, it has been shown experimentally that amyloid-β 42 (Aβ42) and α-synuclein (α-Syn) can coassemble. This has motivated us to investigate the interaction between their monomers as a first step toward exploring the possibility of forming heterodimeric complexes. In particular, our study involves the utilization of various Amber and CHARMM force-fields, employing both implicit and explicit solvent models in replica exchange and conventional simulation modes. This comprehensive approach allowed us to assess the strengths and weaknesses of these solvent models and force fields in comparison to experimental and theoretical findings, ensuring the highest level of robustness. Our investigations revealed that Aβ42 and α-Syn monomers can indeed form stable heterodimers, and the resulting heterodimeric model exhibits stronger interactions compared to the Aβ42 dimer. The binding of α-Syn to Aβ42 reduces the propensity of Aβ42 to adopt fibril-prone conformations and induces significant changes in its conformational properties. Notably, in AMBER-FB15 and CHARMM36m force fields with the use of explicit solvent, the presence of Aβ42 significantly increases the β-content of α-Syn, consistent with the experiments showing that Aβ42 triggers α-Syn aggregation. Our analysis clearly shows that although the use of implicit solvent resulted in too large compactness of monomeric α-Syn, structural properties of monomeric Aβ42 and the heterodimer were preserved in explicit-solvent simulations. We anticipate that our study sheds light on the interaction between α-Syn and Aβ42 proteins, thus providing the atom-level model required to assess the initial stage of aggregation mechanisms related to Alzheimer's and Parkinson's diseases.

3D‐SRDM: 3D super‐resolution of MRI volumes based on diffusion model

AbstractBrain magnetic resonance imaging (MRI) is crucial for diagnosing and understanding neurological disorders, but inherent limitations hinder the visualization of fine details in brain structures. The emergence of super‐resolution techniques, especially deep‐learning methods, has improved imaging quality of MRI, by increasing MRI spatial resolution. At present, deep‐learning algorithms mostly performed super‐resolution on 2D MRI images. However, considering 3D nature of MRI, 3D models are more suitable for brain MRI super‐resolution. To achieve finer brain structural details, this study proposes a 3D brain MRI super‐resolution method based on diffusion model (3D‐SRDM), which is a fast and easily trainable neural network for the generation of high‐resolution brain MRI images. In our 3D‐SRDM model, the self‐attention module in U‐Net is replaced with 3D spatial attention mechanism. The network structure of 3D‐SRDM is optimized to reduce training parameters. Moreover, accelerated sampling from denoising diffusion implicit model is also incorporated to reduce time consumption. By these optimizations, compared with original diffusion model, the proposed model can achieve about 10‐ and 5‐fold speed increase at 4× and 8× super‐resolution of 3D brain MRI volumes, respectively, almost without affecting image quality. Thus, 3D‐SRDM has potential application value in efficiently generating high‐resolution 3D brain MRI images, thus facilitating the doctors' diagnosis.

Comparative analysis of surface phase diagrams in aqueous environment: Implicit vs explicit solvation models.

Identifying the stable surface phases under a given electrochemical conditions serves as the basis for studying the atomistic mechanism of reactions at solid/water interfaces. In this work, we systematically compare the performance of the two main approaches that are used to capture the impact of an aqueous environment, implicit and explicit solvent, on surface energies and phase diagrams. As a model system, we consider the magnesium/water interface with (i) Ca substitution and (ii) proton and hydroxyl adsorption. We show that while the implicit solvent model is computationally very efficient, it suffers from two shortcomings. First, the choice of the implicit solvent parameters significantly influences the energy landscape in the vicinity of the surface. The default parameters benchmarked on solvation in water underestimate the energy of the dissolved Mg ion and lead to spontaneous dissolution of the surface atom, resulting in large differences in the surface energetics. Second, in systems containing a charged surface and a solvated ion, the implicit solvent model may not converge to the energetically stable ionic charge state but remain in a high-energy metastable configuration, representing the neutral charge state of the ion. When these two issues are addressed, surface phase diagrams that closely match the explicit water results can be obtained. This makes the implicit solvent model highly attractive as a computationally-efficient surrogate model to compute surface energies and phase diagrams.

ADVANCED LEARNING AS AN IMPLICIT MODEL OF CONTINUOUS EDUCATION

By modeling and analyzing the levels of knowledge of students in school disciplines, the great didactic significance of more general and universal ideas in the process of forming students’ thinking during learning has been revealed. Presenting particularly important aspects of the process of generalization of students' knowledge and existing interpretations of one model of advanced learning, ways to implement the ideas of continuous education within the framework of fundamentalization and strengthening of the methodological approach of the educational process are identified. The application of the presented model of advanced learning allows to generalize fundamental ideas, universal concepts and methods, which are the basis for the formation of thinking in the following discipline by using general logical methods of scientific knowledge (analysis, synthesis, induction, deduction, modeling, analogues, etc.). The subject of the study is to reveal the fundamental possibilities of advanced learning in the process of implicit transfer of students' knowledge. The purpose of the article is to substantiate the development and implementation of an advanced learning model aimed at improving the quality of learning in the context of continuous education, based on the implicit transfer of student knowledge. It is shown that in advanced learning, the objects of advance are such fundamental ideas, concepts, and methodological approaches which in their essence are core for the process of knowledge transfer. Consistent application of advanced learning is an effective tool for generalizing students' knowledge which contributes to the formation of students' thinking. As a result, the concept of “Advanced learning as an implicit model of continuous education” was has been put forward.

Theoretical derivation of super-ellipse model as approximation of physics-based implicit model for solar PV

Recently, it has been shown that the super-ellipse model can be used to represent the voltage-current characteristics of illuminated solar PV modules. Since the model is claimed to be empirical in nature, this paper demonstrates that the super-ellipse model can be derived as a theoretical approximation of the physics-based implicit model of solar PV. Further, the model parameters used in the super-ellipse model can be approximated to the physical parameters of the implicit model of solar PV. Here, it is also shown how the model parameters of the super-ellipse model vary with the ambient temperature and solar irradiance. The super-ellipse model is compared with other explicit models for different solar modules and shows satisfactory results in representing the voltage-current (current–density) curve of the solar PV module.

Integrating Explicit and Implicit Fullerene Models into UNRES Force Field for Protein Interaction Studies.

Fullerenes, particularly C60, exhibit unique properties that make them promising candidates for various applications, including drug delivery and nanomedicine. However, their interactions with biomolecules, especially proteins, remain not fully understood. This study implements both explicit and implicit C60 models into the UNRES coarse-grained force field, enabling the investigation of fullerene-protein interactions without the need for restraints to stabilize protein structures. The UNRES force field offers computational efficiency, allowing for longer timescale simulations while maintaining accuracy. Five model proteins were studied: FK506 binding protein, HIV-1 protease, intestinal fatty acid binding protein, PCB-binding protein, and hen egg-white lysozyme. Molecular dynamics simulations were performed with and without C60 to assess protein stability and investigate the impact of fullerene interactions. Analysis of contact probabilities reveals distinct interaction patterns for each protein. FK506 binding protein (1FKF) shows specific binding sites, while intestinal fatty acid binding protein (1ICN) and uteroglobin (1UTR) exhibit more generalized interactions. The explicit C60 model shows good agreement with all-atom simulations in predicting protein flexibility, the position of C60 in the binding pocket, and the estimation of effective binding energies. The integration of explicit and implicit C60 models into the UNRES force field, coupled with recent advances in coarse-grained modeling and multiscale approaches, provides a powerful framework for investigating protein-nanoparticle interactions at biologically relevant scales without the need to use restraints stabilizing the protein, thus allowing for large conformational changes to occur. These computational tools, in synergy with experimental techniques, can aid in understanding the mechanisms and consequences of nanoparticle-biomolecule interactions, guiding the design of nanomaterials for biomedical applications.

Diffusion Generative Model-Based Learning for Smart Layer-Wise Monitoring of Additive Manufacturing

Abstract Despite the rapid adoption of deep learning models in additive manufacturing (AM), significant quality assurance challenges continue to persist. This is further emphasized by the limited availability of sample objects for monitoring AM-fabricated builds. Thus, this study advances an emerging diffusion generative model, i.e., the denoising diffusion implicit model (DDIM), for layer-wise image augmentation and monitoring in AM. The generative model can be used to generate potential layer-wise variations, which can be further studied to understand their causation and prevent their occurrence. The proposed models integrate two proposed kernel-based distance metrics into the DDIM framework for effective layer-wise AM image augmentation. These newly proposed metrics include a modified version of the kernel inception distance (m-KID) as well as an integration of m-KID and the inception score (IS) termed KID-IS. These novel integrations demonstrate great potential for maintaining both similarity and consistency in AM layer-wise image augmentation, while simultaneously exploring possible unobserved process variations. In the case study, six different cases based on both metal-based and polymer-based fused filament fabrication (FFF) are examined. The results indicate that both the proposed DDIM/m-KID and DDIM/KID-IS models outperform the four benchmark methods, including the popular denoising diffusion probabilistic models (DDPMs), and three other generative adversarial networks (GANs). Overall, DDIM/KID-IS emerges as the best-performing model with an average KID score of 0.840, m-KID score of 0.1185, peak signal-to-noise ratio (PSNR) of 18.150, and structural similarity index measure (SSIM) of 0.173, which demonstrated strong capabilities in generating potential AM process variations in terms of layer-wise images.

Analyze as randomized—Why dropping block effects in designed experiments is a bad idea

AbstractAgricultural experiments are often laid out as blocked designs such as the randomized complete block designs (RCBD) or split‐plot designs (SPD). Statistical analysis should follow the principle “analyze as randomized.” However, block effects are often not modeled or are typically dropped from the model when non‐significant. Additionally, if linear mixed models are fitted by REML with a non‐negativity constraint on all variances, an implicit model reduction occurs when the variance estimate for random block effects becomes negative in the final iteration. This study investigates the consequences of these types of model reduction on the Type I error rate and the standard error of a treatment difference (s.e.d.) by Monte Carlo simulation for experiments designed as RCBD or SPD. The number of blocks and treatments and the ratio of block or main‐plot error variance to residual error variance were varied, resulting in 27 scenarios. Dropping the block effect by default resulted in deflated Type I error rates and increased the s.e.d. in the RCBD. For the SPD, Type I error rates were inflated for the sub‐plot factor and the main‐plot‐by‐sub‐plot factor interaction, but were deflated for the main‐plot factor. Adverse effects of model reduction were reduced, but did not vanish completely, for model reduction based on significance testing. Implicit model reduction led to inflated Type I error rates for small variance ratios and small datasets. We conclude that block effects as well as main‐plot error effects need to be included by all means in statistical analysis.

Homo anthropologicus: Unexamined behavioural models in sociocultural anthropology

Inferences from ethnography in sociocultural anthropological arguments frequently rely on an unexamined model of the human mind and behaviour. Across a range of theoretical approaches, human thought and behaviour are implicitly understood as coherently following a single underlying cultural logic, described in terms such as ‘ontology’, habitus, political strategy. We term this implicit model Homo anthropologicus, by analogy with Homo economicus. Both simplify human behaviour and can thus lead to errors in its interpretation. We examine examples of Homo anthropologicus in anthropological approaches to ontology, caste, state evasion, and habitus. We propose that such accounts are erroneous in light of the multiple cognitive systems involved in human thought and behaviour, discussed with close reference to dual process theory. Unlike Homo anthropologicus, Homo sapiens’ behaviour is frequently inconsistent. Whilst anthropologists have long acknowledged this is the case, in practice, as we demonstrate through our examples, inconsistency is frequently seen as a problem to be explained away rather than as a feature of behaviour to be accounted for in its own right. We therefore conclude by calling for a greater degree of methodological reflexivity when making inferences from ethnography.

A theoretical study on toluene oxidization by OH radical

Toluene, a prominent member of volatile organic compounds (VOCs), exerts a substantial adverse influence on both human life and the environment. In the context of advanced oxidation processes, the ·OH radical emerges as a highly efficient oxidant, pivotal in the elimination of VOCs. This study employs computational quantum chemistry methods (G4MP2//B3LYP/6-311++G(d,p)) to systematically investigate the degradation of toluene by ·OH radicals in an implicit solvent model, and validates the rationale of choosing a single-reference method using T1 diagnostics. Our results suggest three possible reaction mechanisms for the oxidation of toluene by ·OH: firstly, the phenyl ring undergoes a hydrogen abstraction reaction followed by direct combination with ·OH to form cresol; secondly, ·OH directly adds to the phenyl ring, leading to ring opening; thirdly, oxidation of sidechain to benzoic acid followed by further addition and ring opening. The last two oxidation pathways involve the ring opening of toluene via the addition of ·OH, significantly facilitating the process. Therefore, both pathways are considered feasible for the degradation of toluene. Subsequently, the UV-H2O2 system was designed to induce the formation of ·OH for toluene degradation and to identify the optimal reaction conditions. It was demonstrated that ·OH and 1O2 are the primary active species for degrading toluene, with their contribution ranking as ·OH > 1O2. The intermediates in the mixture solution after reactions were characterized using GC–MS, demonstrating the validity of theoretical predictions. A comparative study of the toluene consumption rate revealed an experimental comprehensive activation energy of 10.33 kJ/mol, which is consistent with the preliminary activation energies obtained via theoretical analysis of these three mechanisms (0.56 kJ/mol to 13.66 kJ/mol), indicating that this theoretical method can provide a theoretical basis for experimental studies on the oxidation of toluene by ·OH.

Unsupervised anomaly detection for radar active deception jamming based on denoising diffusion implicit model

Unsupervised anomaly detection for radar active deception jamming based on denoising diffusion implicit model

A multi-material topology optimization approach to hybrid material structures with gradient lattices

In this paper, we present a novel approach for designing hybrid material structures that incorporate solid, void, and spatially-varying lattices. This approach extends the recursively defined power-law type multi-material interpolation model by incorporating additional gradient material phases. Unlike conventional models, the new extended model considers the secondary ``solid'' material phase as a type of lattice, enabling adjustment of lattice geometry and mechanical behavior with gradient control variables such as density or size. By means of numerical homogenization and interpolation, we explicitly express the effective mechanical behavior of the gradient lattice in relation to the gradient control variable. This approach enhances design freedom by allowing for flexible control of individual material volumes, and also the lower and upper bounds of gradient variables. By using implicit geometrical modeling techniques, the resulting gradient control variable field can be interpreted to create spatially-varying and smoothly-connected lattices at any arbitrary resolution. The effectiveness and flexibility of the proposed approach have been demonstrated through a series of benchmark design tests considering various lattice material types.

ResNet50-1D-CNN: A new lightweight resNet50-One-dimensional convolution neural network transfer learning-based approach for improved intrusion detection in cyber-physical systems

The cyber-physical system (CPS) plays a crucial role in supporting critical infrastructure like water treatment facilities, gas stations, air conditioning components, and smart grids, which are essential to society. However, these systems are facing a growing susceptibility to a wide range of emerging attacks. Cyber-attacks against CPS have the potential to cause disruptions in the accurate sensing and actuation processes, resulting in significant harm to physical entities and posing concerns for the overall safety of society. Unlike common security measures like firewalls and encryption, which often aren't enough to deal with the unique problems that CPS architectures present, deploying machine learning-based intrusion detection systems (IDS) that are specifically made for CPS has become an important way to make them safer. The application of machine learning algorithms has been suggested as a means of mitigating cyber-attacks on CPS. However, the limited availability of labelled data pertaining to emerging attack techniques poses a significant challenge to the accurate detection of such attacks. In the given scenario, transfer learning emerges as a promising methodology for the detection of cyber-attacks, as it involves the implicit modelling of the system. In this research, we propose a new lightweight transfer learning method via ResNet50-CNN1D for intrusion detection in CPS. The Adaptive Gradient (Adagrad) optimizer was applied in the proposed model to minimize the loss function through the adjustment of network weight. We tested how well the suggested ResNet50-1D-CNN model worked using the UNSW-NB15 dataset and a control system dataset called HAI. The HAI dataset was taken from the testbed and based on a planned physical attack scenario. By calculating the coefficient scores for the top ten (10) features in the HAI and UNSW-NB15 data, it was possible to determine the relevance of a feature. The rationale behind employing transfer learning was to mitigate the complexity associated with the classification of cyber-attacks and runtime. The utilization of transfer learning resulted in notable reductions in both the training and testing times required for the detection of attacks. On the HAI data, the results showed an accuracy of 97.32 %, recall of 98.41 %, F1-score of 96.32 %, and precision of 97.09 %. On the UNSW-NB15 data, the results showed an accuracy of 99.89 %, recall of 99.09 %, F1-score of 98.01 %, and precision of 98.70 %.

An indirect training approach for implicit constitutive modelling using recurrent neural networks and the virtual fields method

Accurate material description is crucial to achieve high-quality results in computational analysis software. Phenomenological constitutive laws generalize the material behaviour observed in simple mechanical tests. The resulting empirical expressions contain parameters that need to be calibrated through an inverse optimization process. Advancements in Digital Image Correlation (DIC) techniques have enabled the extraction of non-uniform multi-axial displacement fields, facilitating the development of heterogeneous mechanical specimens and unlocking access to richer material behaviour data. Despite these advancements, constitutive models are still limited by mathematical formulations and biases due to simplifying assumptions. Artificial Neural Networks (ANNs), as universal function approximators, could potentially drive a paradigm shift in the field. ANNs do not require explicit pre-formulations, hence avoiding the need for identifying unknown parameters. Moreover, ANNs are able to implicitly learn patterns purely from data, allowing to circumvent the biases induced by the simplifying assumptions of first-order principles. However, the training of ANNs for implicit constitutive modelling is not straightforward, given the impossibility of obtaining direct measurements of stress. This paper explores the integration of Recurrent Neural Networks (RNNs) with the Virtual Fields Method (VFM) for material modelling. This approach uses global force and displacement data to indirectly train the neural network, with the equilibrium being evaluated globally through the VFM loss function. The proposed method is (i) computationally efficient by not requiring Finite Element Analysis (FEA), (ii) compatible with both full-field measurements and numerically generated data, and (iii) able to handle experimental boundary conditions.

Consistent Analytical Second Derivatives of the Kohn-Sham DFT Energy in the Framework of the Conductor-Like Screening Model through Gaussian Charge Distributions.

The use of implicit solvation models such as the conductor-like screening model (COSMO) in quantum chemical calculations is very common, as both a rough estimate of solvation effects as well as a general tool for stabilizing ionic molecular structures. In order to generate a smooth potential energy surface as well as consistent gradients, it is necessary to apply the Gaussian charge model (GCM) for the COSMO charges. This work introduces an efficient implementation for consistent analytical second derivatives of the electronic energy with COSMO-GCM in the framework of the Kohn-Sham density functional theory. This is used to investigate the infrared spectroscopy of amino acids in aqueous solution, where the impact of pH on the molecular structure and vibrational spectra is examined. Furthermore, the structure and stability of selected all-metal aromatic cluster ions are assessed.

3D geometrical modelling of the non-cylindrical Vélodrome Miocene fold in the southwestern Alps

Fold geometries and kinematics within foreland basins is a major issue for understanding the late evolution of thrust fronts. In the foreland of the southwestern Alps, the Vélodrome complex fold involves the whole Tertiary series which have recorded the evolution of the alpine front. From a geometrical and kinematic point of view, the Vélodrome is classically described as a recumbent Mio-Pliocene syncline with a strongly curved axis; interpreted either as a classic growth-fold, a post-deposit fold, or a result of coeval salt activity. The debate reveals a lack of consensus on the internal structure of the Vélodrome and the potential role of salt motion. By a detailed field analysis coupled with an implicit 3D geometrical modelling approach, we test the different hypotheses and provide a new 3D illustrated interpretation of the Vélodrome series. The Vélodrome consists in a complex non-cylindrical structure composed by several folds with different orientations together with inter- and intra-formations unconformities. Folding is partly syn-depositional and began earlier in the south, near Esclangon village, than in the north. In the north, deformation started with the marly and sandy molasse (m2, middle Burdigalian), attested by the transition from the conformable conglomeratic marine molasse (m1–2, Aquitanian and early Burdigalian) to the unconformable m2 to the Valensole formation. Deformation began earlier in the south, during the deposition of the m1–2, as shown by internal unconformities. We discuss the ingredients controlling the deformation in the Vélodrome, regional tectonics (mainly since middle Burdigalian) versus salt tectonics. This study further brings new constraints on the timing and pattern of deformation of the southwestern orogenic front of the Alps. It also highlights the power of the 3D geometrical modelling approach for testing different hypotheses and better understanding 3D complex structures.

GMMFormer: Gaussian-Mixture-Model Based Transformer for Efficient Partially Relevant Video Retrieval

Given a text query, partially relevant video retrieval (PRVR) seeks to find untrimmed videos containing pertinent moments in a database. For PRVR, clip modeling is essential to capture the partial relationship between texts and videos. Current PRVR methods adopt scanning-based clip construction to achieve explicit clip modeling, which is information-redundant and requires a large storage overhead. To solve the efficiency problem of PRVR methods, this paper proposes GMMFormer, a Gaussian-Mixture-Model based Transformer which models clip representations implicitly. During frame interactions, we incorporate Gaussian-Mixture-Model constraints to focus each frame on its adjacent frames instead of the whole video. Then generated representations will contain multi-scale clip information, achieving implicit clip modeling. In addition, PRVR methods ignore semantic differences between text queries relevant to the same video, leading to a sparse embedding space. We propose a query diverse loss to distinguish these text queries, making the embedding space more intensive and contain more semantic information. Extensive experiments on three large-scale video datasets (i.e., TVR, ActivityNet Captions, and Charades-STA) demonstrate the superiority and efficiency of GMMFormer.

Boosting Residual Networks with Group Knowledge

Recent research understands the residual networks from a new perspective of the implicit ensemble model. From this view, previous methods such as stochastic depth and stimulative training have further improved the performance of the residual network by sampling and training of its subnets. However, they both use the same supervision for all subnets of different capacities and neglect the valuable knowledge generated by subnets during training. In this manuscript, we mitigate the significant knowledge distillation gap caused by using the same kind of supervision and advocate leveraging the subnets to provide diverse knowledge. Based on this motivation, we propose a group knowledge based training framework for boosting the performance of residual networks. Specifically, we implicitly divide all subnets into hierarchical groups by subnet-in-subnet sampling, aggregate the knowledge of different subnets in each group during training, and exploit upper-level group knowledge to supervise lower-level subnet group. Meanwhile, we also develop a subnet sampling strategy that naturally samples larger subnets, which are found to be more helpful than smaller subnets in boosting performance for hierarchical groups. Compared with typical subnet training and other methods, our method achieves the best efficiency and performance trade-offs on multiple datasets and network structures. The code is at https://github.com/tsj-001/AAAI24-GKT.

Lyapunov-Stable Deep Equilibrium Models

Deep equilibrium (DEQ) models have emerged as a promising class of implicit layer models, which abandon traditional depth by solving for the fixed points of a single nonlinear layer. Despite their success, the stability of the fixed points for these models remains poorly understood. By considering DEQ models as nonlinear dynamic systems, we propose a robust DEQ model named LyaDEQ with guaranteed provable stability via Lyapunov theory. The crux of our method is ensuring the Lyapunov stability of the DEQ model's fixed points, which enables the proposed model to resist minor initial perturbations. To avoid poor adversarial defense due to Lyapunov-stable fixed points being located near each other, we orthogonalize the layers after the Lyapunov stability module to separate different fixed points. We evaluate LyaDEQ models under well-known adversarial attacks, and experimental results demonstrate significant improvement in robustness. Furthermore, we show that the LyaDEQ model can be combined with other defense methods, such as adversarial training, to achieve even better adversarial robustness.