Physics Constrained Flow Neural Network for Millisecond-timescale Predictions in Data Communication Networks

Data communication networks provide the majority of data transmission services to support today’s internet applications and present huge social value. This paper proposes a dynamic analysis model to deal with flows of information from sources to destinations in data communication networks. As depicted in Fig. 1, the source of a flow is a node (computer, phone, router/switch, etc.) in the network from which packets¹¹1Information is encapsulated in packets, which can be seen as particles travelling in the network. start their travel and the destination is where they end. Throughout the lifespan of a flow (e.g., a 10-minute phone call or a 2-hour online video), packets consistently travel along the assigned routing path connecting the source and destination. Depending on the real-time congestion conditions of nodes on the routing path, packets may experience different buffering or retransmission delays, resulting in varying transmission rates and service delays.

A good understanding of the behavioral patterns of the underlying packets and flows plays a crucial role in network planning, traffic management, as well as optimizing Quality of Service (QoS, e.g., transmission rate, delay, etc.). However, the highly nonlinear dynamics of the networking system often create challenges for a fine-grained analysis. Particularly, at the milisecond timescale, the traffic traces are highly skewed and it is difficult to recognize obvious patterns.

Therefore, recent works often rely on off-the-shelf machine learning models to make predictions based on the data statistics of a flow. The remarkable advances achieved in long-timescale prediction tasks (e.g., at seconds or minutes intervals Marcus et al. (2020); Mozo et al. (2018)) are a testament to this minimalist principle. A key limitation of these machine learning models is that they disregard the physics governing the generative process of these packet flows.

In practice, the dynamics of spatio-temporal flows are governed by physical laws or models regulating the networking system. For instance, in networks with a TCP/IP protocol suite, the real-time transmission rates of different packet sources are typically modulated dynamically by congestion control models to prevent congestive collapse Nathan et al. (2019). The conservation law describes the macroscopic fluid dynamics of data flows during hop-by-hop transmissions Apice et al. (2008). The physics establishes an inductive bias Peter et al. (2018) when predicting from history. That is, the future is not entirely random or noisy even when observed at a millisecond granularity, but can be predicted from physical models or laws.

As illustrated in Fig. 1, assume $1MB$ of data is sent out from the source node $n_{0}$ (i.e., Sender1) at the first time interval $t=1ms$. After a certain delay (e.g., $1ms$) due to link propagation and packet processing, the first $1MB$ of data will arrive at its next routing node $R_{1}$ at $t=2ms$. In this case, at $t=1ms$, the sending rate at the source node $n_{0}$ and the receiving rate at router $R_{1}$ are $x_{r,n_{0}}^{t_{1}}=1Gbps$ and $x_{r,R_{1}}^{t_{1}}=0Gbps$, respectively. At $t=2ms$, the associated rate at $R_{1}$ becomes $x_{r,R_{1}}^{t_{2}}=1Gbps$, while $x_{r,n_{0}}^{t_{2}}$ is again determined by the congestion control model according to the acknowledged ACK information. Such forwarding process will continue hop by hop until reaching the destination (Receiver1).

The above hop-by-hop data transmission establishes a special connection between space and time. We can observe in Fig. 1 that $x_{r,R_{1}}^{t_{2}}=1Gbps$ is physically determined by $x_{r,n_{0}}^{t_{1}}=1Gbps$, the history information of its predecessor node (i.e, source $n_{0}$) rather than its own history record $x_{r,R_{1}}^{t_{1}}=0Gbps$. Therefore, such time-resolved flow data not only tells us who is related to whom, but also when and in which order relations occur. This provides a specific relational bias for modelling the spatio-temporal evolution of such flow data.

In this paper, we investigate how to embed such essential physical bias in a learning model to improve the short-timescale (millisecond-level) flow predictions. Accordingly, we propose Flow Neural Network (FlowNN), the first customized learning model for the data analysis of networking flows, while respecting packet routing structures and network mechanisms.

Our contributions can be summarized as follows: (1) We develop FlowNN, a prediction model that exploits data correlations physically induced by the congestion control model and flow conservation rule during hop-by-hop data forwarding; (2) We design a self-supervised learning strategy with stop-gradient that can learn the feature representation and physical bias simultaneously at each gradient update step; (3) We show with realistic packet datasets that FlowNN achieves consistently better results and outperforms the incumbent baselines by reducing the loss by 17% $\sim$ 71% for short-timescale prediction problems.

Packet Flow. An $s$-$d$ flow is a sequence of packets encapsulating the message bits and getting forwarded from source $s$ along an assigned routing path $P\in\mathbb{Z}^{L}$ to destination $d$. Let $x_{f,n}^{t}\in\mathbb{R}$ be the value of feature $f$ measured at node $n\in P$ during the time interval $t$. For instance, $x_{f,n}^{t}$ can be the average packet travelling delay or transmission/receiving rate of a flow. Let us denote the time-series tensor of a flow as $X^{t}=\{x_{n}^{t}\}_{n=1,\cdots,L}\in\mathbb{R}^{L\times N_{f}}$. With telemetry monitors installed at each node, we can collect the running traces of $X^{t}$ as shown in Fig. 2(a).

Eq. 1 analytically constrains the sending and receiving bit rates at each node along the routing path. As a consequence, without congestion, the bit rates observed at each routing node will be exactly the same. Nevertheless, if routing nodes experience different congestion due to link intersection, the observed rates between two neighboring routing nodes will become larger and smaller over time during the flow lifespan. As shown in Fig. 2(b), this explicitly forms a set of paired source-target time windows within the time series of two neighboring nodes. During the source window $T_{1}$, a node (e.g., node 1) presents larger rates than its successor node (node 2). Then, in the target window $T_{2}$, node 1 will present smaller rates than node 2. Finally, the cumulative amount of forwarded flow bits are conserved among the routing nodes. Such property arises from the routing structure and packet buffering mechanism in communications networks, and is independent of network and flow configurations.

Accurately modelling the flow dynamics requires attention to the above domain-specific physics and properties, which are absent in most (if not all) existing learning models.

Next, we introduce FlowNN to embed the physics as a learning bias to improve data analysis.

Our architecture (Fig. 3) consists of three learnable neural network functions, denoted as $f_{L_{1}}$, $f_{L_{2}}$, $f_{L_{3}}$, respectively. FlowNN works upon the initial embedding layer $L_{1}$ to further impose the learning bias from the working mechanism of physical systems.

Concretely, for the time-series tensor of each flow $X^{t}=\{x_{n}^{t}\}_{n=1,\cdots,L}\in\mathbb{R}^{L\times N_{f}}$, $f_{L_{1}}$ use a MLP to project each $x_{n}^{t}\in\mathbb{R}^{N_{f}}$ to latent space $\mathbb{R}^{d}$:

Then, PathAggregator $f_{L_{2}}$ uses a MLP to aggregate $H^{\tau}=\{h_{n}^{\tau}\}_{n=1,\cdots,L}\in\mathbb{R}^{L\times d}$ from layer $L_{1}$ along dimension-$n$ and a recurrent net (e.g., GRU) to aggregate over time $\tau$:

The output of PathAggregator, $\hat{h}_{1}^{\tau}$, aggregates the locally-perceived traffic and network conditions along the routing path, which can be mapped to its next sending rate from the source node. It physically plays the role of a simulator to imitate the decision process of existing congestion control models.

With the aggregated path state $\hat{h}_{1}^{t}$ and the local hidden states of individual nodes, $\{h_{n}^{t}\}_{n=2,\cdots,L}$ from $f_{L_{1}}$, the induction layer $L_{3}$ is then applied to estimate the new local states of each node under the new sending state (i.e., $\hat{h}_{1}^{\tau}$). Specifically, for each pair³³3$\langle\cdot\rangle$ denotes the paired set of the two included elements. of two neighboring nodes $\langle n,n+1\rangle$, $n=1,\cdots,L-1$, we split the associated time-series $\hat{h}_{n}^{\tau}$ into a set of correlation windows $\langle T_{1}^{i},T_{2}^{i}\rangle_{i=1,2,\cdots}$ such that the boundaries of each time window correspond to timestamps where the rate difference between two neighboring nodes is 0, i.e., $x_{r,n}^{\tau}-x_{r,n+1}^{\tau}=0$ (see Fig. 2(b)). For each correlated window $\langle T_{1}^{i},T_{2}^{i}\rangle$, we encode the hidden states of sequences in $T_{1}^{i}$ and $T_{2}^{i}$ as follows:

where $Seq2Seq$ is a sequence to sequence net Sutskever et al. (2014) to predict the target sequences $\{\hat{h}_{n+1}^{\tau}\}_{\tau\in T_{2}^{i}}$ conditioned on the states of source sequences $\{\hat{h}_{1}^{t}\}_{t\in T_{1}^{i}}$. In the experiments, we used both GRU-type encoders and decoders to implement the $Seq2Seq$ net. Here, $||$ and $|$ are concatenation and conditional operators, respectively.

Eq. 4 introduces path-level messages (i.e., $\hat{h}_{1}^{\tau}$) into the locally-perceived states of node $n+1$ for all $\tau\in T_{1}^{i}$. Eq. 5 predicts the states of sequences in $T_{2}^{i}$ at $n+1$ conditioned on the states of history window $T_{1}^{i}$.

The steps from Eq. 3–5 explicitly embed the natural data connections in Fig. 2(a) arising from the congestion control and hop-by-hop data transmission. As shown in Fig. 3, the predicted outputs $\{\hat{h}_{n}^{t}\}_{n=1,\cdots,L}$ can be recurrently fed into the same PathAggregator $L_{2}$ and Induction $L_{3}$ to propagate messages by the rules of Eq. 3–5 to farther distances. The number $N$ of recurrent iterations is a tunable hyperparameter.

The self-supervised FlowNN training architecture (Fig. 8) has two branches, working on a shared initial embedding layer (encoder) $f_{L_{1}}$. The bottom branch predicts the states at time $t$ based on the information up to time $t-1$ by using $f_{p}$ (i.e., PathAggregator $L_{2}$ and Induction $L_{3}$) to impose the physical bias upon the outputs of $f_{L_{1}}$. The top branch embeds the ground truth features at time $t$ with a MLP projection $f_{m}$ that transforms the outputs from $f_{L_{1}}$. Denote the output vectors of the two branches as $\mathcal{Z}=(f_{m}(h_{n}^{t}))_{n=1,\cdots,L}$ and $\hat{\mathcal{Z}}=\{\hat{h}_{n}^{t}\}_{n=1,\cdots,L}$. Similar to Grill et al. (2020); Chen and He (2021), we define the following symmetrized loss to minimize their negative cosine similarity:

where the $\mathrm{stopgrad}(\hat{\mathcal{Z}})$ operation detaches $\hat{\mathcal{Z}}$ from the gradient computation, and thus $f_{p}$ receives no gradient from the first term of loss $\mathcal{L}$ (and vice versa for $\mathcal{Z}$).

The training strategy in Fig. 8 improves Contrastive Predictive Coding Oord et al. (2019) with a stop-gradient operation. As annotated in Fig. 8, the output of the bottom pipeline is the predicted hidden vectors (the information at $t$ is NOT included when predicting for time $t$ with $f_{p}$). If we first remove the projector $f_{m}$, the upper pipeline produces the true encoded hidden vectors from $f_{L_{1}}$ with the information at $t$ included (thus this can be treated as the ground truth of hidden vectors in latent space). The loss here is constructed to make the predicted hidden vectors agree with the true encoded hidden vectors. The pretraining process tries to learn with FlowNN the physical bias arising from the built-in mechanisms of networks. Here, the $f_{m}$ is added to approximate the expectation of all possible inputs from each given input sample. This makes the learned physical bias from FlowNN stable across different inputs. This precisely explains the good transferability of the pretrained FlowNN in cross-task tests (Table 3) and Out-Of-Distribution tests (Table 4). Refer to Appendix C for more details.

We perform experiments to predict the end-to-end packet transmission delay and sending/receiving rates $x_{r,n}^{t+1}$ along the routing path of each individual flow. The predictions of these two features can provide critical information for the transmission optimization for each service.

The compared baselines treat the time-series tensor of a flow as multivariate input (GRU, m-GRU) or graph-type input (STHGCN, STGCN), which are the dominant approaches to model the spatio-temporal relationships in the existing studies. Note that FlowNN reduces to m-GRU model if we remove the induction layer $L_{3}$ and set the number of iterations to $N=1$ in Fig. 3. Therefore, we did not perform further ablation studies in the following experiments. We conducted a grid search over the hyper-parameters for all baselines. The recurrent iteration $N=2$ was chosen. Experiments were conducted on a single Nvidia P40 24GB GPU. More details about the model implementations and hyper-parameter search procedures are given in Appendix E.

Traditional analytical models. The past decades have seen numerous packet analysis models proposed to mathematically model the network dynamics Gebali (2015). For example, extensive studies use the Poisson model to characterize the traffic by assuming that the arrival pattern between two successive packets follows a Poisson process. Considering the heavy tailed distribution and burstiness of the data-center traffic, recent work in Benson et al. (2010) models the packet arrival pattern as a log-normal process. To capture the temporal patterns and make predictions accordingly, ARIMA is exploited in Ergenc (2019) to model the time-series packet traffic. These analytical models often make important assumptions. Moreover, these statistical models work at coarse timescales (e.g., hours) and assume relatively smoother traffic patterns. However, many tasks require the flow statistics at a subsecond timescale, e.g., packet forwarding/queueing configurations and real-time networking control. This implies that tasks requiring analysis models at finer-grained time scales are beyond the capability of these traditional models.

Similar to existing learning models, FlowNN uses a finetuning stage to adapt a pretrained model to various downstream tasks or environments. Nevertheless, the self-supervised learning strategy used by FlowNN is able to embed the physical bias that is universally transferable, provided that the new environment follows the same working mechanisms. The patterns manifested by the data itself may vary in different environments, but the underlying physical working mechanisms typically remain invariant as long as the new environment follows the same principles, e.g., TCP/IP.

In this paper, we developed a customized neural network–FlowNN with the associated self-supervised training strategy to improve the short-timescale prediction problem of information flows in communication networks. This study pioneers the design of a customized neural network and learning strategy by combining network structure and working mechanisms. We reported state-of-the-art performances for multiple practical networking applications, which demonstrates the strength of our approach. As a new ‘playground’ for both networking and deep learning communities, the research of network intelligence is still in its infancy. We hope this work will inspire more innovation in this field in the future.