However,inpractice, we did not find both effects inthe training. The last term in the loss function of COMET does not need the information from the dataset. This section contains the experiment details for cases tested in section 4. In section 4, there are 6 simple experiments performed todemonstrate the capability ofCOMET:(1) mass-spring, (2) 2D Case5: 2Dnonlinear spring.Weconsider acaseofamotion ofanobjectofmassm=1in2D where it is connected to the origin with a nonlinear spring with forceF = |r|2r where r is the position of the object in 2D coordinate. The constants of motion of this systems are the energy and the angular momentum,whichmakesnc=2. Case 6: Lotka-Volterraequation isanordinary differential equation modelling thepopulation of predatorandprey. The training loss in this case was composed of the reconstruction loss and the dynamics loss.

However,ithas attracted less attention when the objects are difficult to represent in astandard way,for example cars or food.

In contrast, the mathematics needed to analyze such schemes is what forms the focus in Stochastic Approximation (SA) theory [2, 4]. More generally, SA refers to an iterative scheme that helps find zeroes or optimal points of a function, for which only noisy evaluationsarepossible.

Because a flow is defined in a coordinate-free form, the Hamilton equation can be defined in a coordinate-free manner as well. For further information, see, e.g., [3, 18, 20]. Differential 2-forms are generally represented by using a skew matrix in the same way. A differential form ω is closed if dω = 0 . Definition 3. A symplectic 2-form is a closed and non-degenerate differential 2-form.

Similarly the ISOMAP face database consists ofimages (256levels ofgray)ofsize64 64,i.e.,vectors in R4096, whereas the correct intrinsic dimension is only3 (for the vertical, horizontal pause and lightingdirection). The second approach, is anaverage caseapproach (in the spirit of thestatistical mechanics treatment ofhighdimensional systems), thatmodelsfeaturevectorsby arandom ensemble,taken as aset ofrandom vectors with independently identically distributed (i.i.d.) components, and a small but xed fraction of non-zero components.

Similarly the ISOMAP face database consists ofimages (256levels ofgray)ofsize64 64,i.e.,vectors in R4096, whereas the correct intrinsic dimension is only3 (for the vertical, horizontal pause and lightingdirection). The second approach, is anaverage caseapproach (in the spirit of thestatistical mechanics treatment ofhighdimensional systems), thatmodelsfeaturevectorsby arandom ensemble,taken as aset ofrandom vectors with independently identically distributed (i.i.d.) components, and a small but xed fraction of non-zero components.

LetT be the time horizon andPT be the path-length that essentially reflects the non-stationarity of environments, the state-of-the-art dynamicregretis O( p T(1+PT)).

Supply chain management faces significant challenges, including demand fluctuations, inventory imbalances, and amplified upstream order variability due to the bullwhip effect. Traditional methods, such as simple moving averages, struggle to address dynamic market conditions. Emerging machine learning techniques, including LSTM, reinforcement learning, and XGBoost, offer potential solutions but are limited by computational complexity, training inefficiencies, or constraints in time-series modeling. Liquid Neural Networks, inspired by dynamic biological systems, present a promising alternative due to their adaptability, low computational cost, and robustness to noise, making them suitable for real-time decision-making and edge computing. Despite their success in applications like autonomous vehicles and medical monitoring, their potential in supply chain optimization remains underexplored. This study introduces a hybrid LNN and XGBoost model to optimize ordering strategies in multi-tier supply chains. By leveraging LNN's dynamic feature extraction and XGBoost's global optimization capabilities, the model aims to mitigate the bullwhip effect and enhance cumulative profitability. The research investigates how local and global synergies within the hybrid framework address the dual demands of adaptability and efficiency in SCM. The proposed approach fills a critical gap in existing methodologies, offering an innovative solution for dynamic and efficient supply chain management.

This review aims to conduct a comparative analysis of liquid neural networks (LNNs) and traditional recurrent neural networks (RNNs) and their variants, such as long short-term memory networks (LSTMs) and gated recurrent units (GRUs). The core dimensions of the analysis include model accuracy, memory efficiency, and generalization ability. By systematically reviewing existing research, this paper explores the basic principles, mathematical models, key characteristics, and inherent challenges of these neural network architectures in processing sequential data. Research findings reveal that LNN, as an emerging, biologically inspired, continuous-time dynamic neural network, demonstrates significant potential in handling noisy, non-stationary data, and achieving out-of-distribution (OOD) generalization. Additionally, some LNN variants outperform traditional RNN in terms of parameter efficiency and computational speed. However, RNN remains a cornerstone in sequence modeling due to its mature ecosystem and successful applications across various tasks. This review identifies the commonalities and differences between LNNs and RNNs, summarizes their respective shortcomings and challenges, and points out valuable directions for future research, particularly emphasizing the importance of improving the scalability of LNNs to promote their application in broader and more complex scenarios.

Because a flow is defined in a coordinate-free form, the Hamilton equation can be defined in a coordinate-free manner as well. For further information, see, e.g., [3, 18, 20]. Differential 2-forms are generally represented by using a skew matrix in the same way. A differential form ω is closed if dω = 0 . Definition 3. A symplectic 2-form is a closed and non-degenerate differential 2-form.