Machine learning aided understanding and manipulating thermal transport in amorphous networks

The thermal transport property of a material is a crucial factor determining the performance and reliability of various devices’ applications, including thermal management¹ and thermoelectric conversion.² A high thermal conductivity is preferred in electronics, as it minimizes temperature rise during device operation, thereby enhancing device performance and lifespan.² Conversely, a low thermal conductivity is valuable in thermoelectric materials, as it gives rise to substantial temperature gradients, a crucial factor for the effective functioning of such devices.³ Amorphous networks are prevalent in various systems, including biological networks,^4,5 dynamical networks,⁶ polymers,^7,8 and nanotube/nanowires,^9,10 and they exhibit great potential of flexible manipulation of its thermal transport properties.¹¹ 

Understanding thermal transport in these networks is crucial for optimizing heat dissipation, designing efficient materials, and comprehending the behavior of amorphous systems.^12–15 However, the relationship between amorphous network structures and their thermal transport property is still an open question.¹⁶ The thermal transport properties of such systems are influenced by a variety of network characteristics, such as node masses,¹⁷ average degree,¹⁸ assortativity coefficient,^19,20 clustering coefficient,²¹ small-worldness,²² and closeness centrality.²³ It is also due to the complicated topological patterns of amorphous networks that we lack a sufficient, representative dataset to investigate the thermal transport of network systems.²⁴ 

To unveil the fundamental physical mechanism dictating heat transport within amorphous networks, a substantial number of network structures serving as the dataset are imperative. By scrutinizing these amorphous structures, one can delve into the inherent thermal transport mechanisms at play. To establish a comprehensive dataset, it is essential to employ a versatile method for modifying the network structure. One common approach altering the topological configuration is achieved by selectively removing links.^25–28 However, this method is constrained to reducing thermal conductance as cutting bonds impedes thermal transport pathways, leading to a decrease in the thermal conductance. An alternative tool is the rewire method,^29,30 which introduces bond crossing areas within the network. Another promising method entails perturbing nodes within amorphous networks toward nearby positions, followed by the application of Delaunay triangulation to update the topological patterns. This particular method has exhibited potential in both augmenting and diminishing thermal conductance. Nevertheless, despite its promise, the tuning procedure lacks an efficient mechanism for guiding node distortion. Improving the guidance efficiency of node distortion remains a pivotal aspect in refining the tuning process for generating amorphous networks with significantly increased or decreased thermal conductance. Machine learning and optimization algorithms are promising to overcome the design impedance in materials research.³¹ 

In this study, we employed the simulated annealing (SA) algorithm to systematically generate a comprehensive dataset of amorphous networks, encompassing configurations that both enhance and reduce thermal conductance. This versatile tuning methodology was applied across networks of diverse sizes, ranging from $16×16$ to $32×32$⁠. The study delves into the generated amorphous networks with a variety of thermal conductance and tries to discover the fundamental physics governing thermal transport within these amorphous networks. Throughout the tuning process, the networks consistently maintained their amorphous state, as indicated by the order parameter $ϕ 6$⁠, which consistently hovered around 0.4. Networks with enhanced thermal conductance exhibited bonds predominantly parallel to the heat flow direction, while networks with reduced thermal conductance showed a tendency for bonds to orient perpendicular to the heat flow direction. This observed orientation tendency is quantified through the average shortest path distance connecting heat source and sink nodes: a reduction in the average path distance from high-temperature nodes to low-temperature nodes enhances thermal transport, whereas an increase in the effective path length impedes thermal transport along the direction from high-temperature to low-temperature nodes.

We choose the random networks for investigation due to their completely stochastic structures, anticipating that the underlying thermal transport mechanisms within such a system would be more universally applicable to various amorphous network structures. Initially, a set of nodes is randomly distributed within a square region with a side length of $N$⁠. Along each boundary of the square region, we randomly positioned $N$ nodes, ensuring that the total number of nodes for each network equals to $N t o t=N×N$⁠, as illustrated in Fig. 1(a). Subsequently, bonds connecting these nodes are established through Delaunay triangulation, resulting in the formation of an amorphous network with identical degree of connectivity, as depicted in Fig. 1(b).

To manipulate the configuration of the random network, we applied a combination of the node distortion method and Delaunay triangulation, generating multiple random networks with varying thermal conductance values to form a dataset. A specific node, denoted by the red circle in Fig. 1(b), is initially displaced to a nearby position. Subsequently, Delaunay triangulation is reapplied, yielding a distinct random network pattern as shown in Fig. 1(c). Notably, the local structure highlighted by the red square in Figs. 1(b) and 1(c) is entirely different. This configuration change resulted in a modified thermal conductance value, transitioning from 2.02 to 1.96. We have observed that manipulating a specific node within the network has the potential to either enhance or diminish the overall conductance of the amorphous network. However, achieving the global optimal values of thermal conductance necessitates ensuring that the node distortion consistently aligns with the trend of continuously increasing or decreasing $C$⁠. Therefore, a generalized approach is required to monitor whether the manipulation of node distortion can be accepted during the tuning process. This oversight is facilitated by the simulated annealing (SA) algorithm.^32–34 

We ignore the impact from contact resistance and only focus on the influence from the network topology to obtain the temperature distribution and heat flux among the amorphous networks. In this case, the topology of an amorphous network can be expressed as an adjacency matrix $A$⁠, with its element $A i j=1$ if nodes $i$ and $j$ are connected and 0 otherwise. Define the $A i i=0$ for avoiding self-connection. Denote $F$ as the temperature field vector of $N t o t$ components. We consider that there are $m$ modes with temperature $F h$ as the heat source and $n$ nodes with temperature $F l$ as the heat sink.

The second equation can be expressed as $L 22 F 2=− L 21 F 1$⁠. As $F 1$ and the elements of all submatrices $L 11$⁠, $L 12$⁠, $L 21$⁠, and $L 22$ are known for a network, so $− L 21 F 1$ is a (⁠ $N t o t−m−n$⁠)-dimensional vector and the equation $L 22 F 2=− L 21 F 1$ has a unique nonzero solution $F 2$⁠. Then, substituting $F 2$ into the first equation, we can obtain $J 0$⁠. Thus, $J 0$ can be analytically solved. Finally, we can calculate the thermal conductance $C$ of the entire amorphous network as $C= J 0/( T h− T l)$⁠. Note that these quantities are dimensionless.

By employing the node distortion method in conjunction with Delaunay triangulation, we can systematically mold the initially random network to attain specific configurations characterized by both enhanced and decreased thermal conductance. The initial configuration, delineated in Fig. 2(a), represents the temperature field at steady state with the color on each node marked the temperature distribution. Nodes along the bottom boundary are applied with high temperatures, and those along the top boundary are set with low temperatures. Therefore, the setting of boundary conditions will cause the overall heat flow to flow from the bottom to the top. In Fig. 2(b), the network configuration illustrates an augmentation in thermal conductance, whereas Fig. 2(c) showcases the ultimate adjusted network with reduced thermal conductance. Intriguingly, the network with enhanced thermal conductance exhibits a prevalence of bonds oriented along the heat flow direction, while the random network with diminished thermal conductance showcases numerous bonds aligned perpendicularly to the flow direction. This observation emphasizes the important role of bonds oriented along the direction of heat flow in increasing the thermal conductivity of the network. Conversely, bonds oriented perpendicular to the direction of heat flow impede heat transport from high-temperature nodes to low-temperature nodes.

The amorphous networks maintained their amorphous state throughout the tuning process, as quantified by the order parameter $ϕ 6$ illustrated in Fig. 2(d). The local hexagonal order parameter for a given node $j$ is quantified by $ψ 6(j)= 1 N j ∑ n = 1 N j=exp(i6 θ n j)$⁠, where $N j$ is the number of $n$ nearest neighbors of $j$ and $θ n j$ is the angle between the horizontal axis and the bond linking $j$ to $n$⁠.³⁵ We considered the average of the absolute values of local the hexagonal order parameter from each node inside the amorphous network as $ϕ 6=⟨| ψ 6(j)|⟩$⁠. In a perfect hexagonal lattice, $ϕ 6=1$⁠, while values of $ϕ 6=0.5$ or less indicate an amorphous structure.³⁶  Figure 2(d) depicts the variations in the order parameter as thermal conductance increased and decreased, respectively, with different network sizes ranging from $16×16$ to $32×32$⁠. Notably, throughout the tuning process, the order parameter consistently hovered around 0.4, a value below the critical threshold of 0.5, indicating that the amorphous networks remained in an amorphous state during tuning.

Another order parameter, clustering coefficient $E$⁠, quantifies the heterogeneity of the amorphous network.^37,38 It is defined as $E= 1 N t o t∑ 2 δ i k i ( k i − 1 )$⁠, where $δ i$ is the number of bonds connected surrounding the node $i$⁠, and $k i$ is the number of neighbor nodes around $i$⁠, which is the degree of $i$⁠. $E$ represents the probability that two neighbors of a node are also connected to each other. For larger $E$⁠, the links form groups and the network is more likely to generate local clusters. The heterogeneity also influences the heat flux through the network. As shown in Fig. 2(c), the clustering coefficient for networks tuned to enhanced and decreased thermal conductance values is kept at around 0.1, independent with the network sizes from $16×16$ to $32×32$⁠. This indicates that the inherent thermal transport tailored by our method is not dependent on the network clustering coefficient.

The average shortest path length is a critical property in the general analysis of amorphous networks. It refers to the total length of bonds connecting any two nodes within a network, representing the shortest route from a source node to a target node. The average shortest distance, denoted as $L$⁠, is influenced by both network size and structural characteristics of the network, impacting heat conduction dynamics.^39–41 

In thermal transport, a shorter distance between the heat source and sink nodes promotes more efficient heat transfer, while a longer transport distance indicates inefficiency in guiding heat in the desired direction. We have observed that thermal conductance is inversely proportional to the average shortest distance between connecting edges of hot and cold nodes. For analysis, we define pairs of heat source and sink nodes located on the bottom and upper boundaries, sharing the same $x$ coordinate, as $p$ (where $p=1,2,…,N$ for N nodes on each boundary). The Floyd algorithm^42,43 is employed to find all edges generating the shortest path distance $l p$ between each source and sink node pair. As illustrated in Fig. 3(a), the left panel represents a configuration of the maximized thermal conductance, while the right panel represents a configuration of the minimized thermal conductance. Both configurations are derived from the same original amorphous network using our node displacement and Delaunay triangulation method.

The disparity in the shortest distance $l p$ between the left and right panels is apparent. On the left panel, the red curve forms an almost straight line connecting the selected pair of source and sink nodes, resulting in a smaller $l p$⁠. Conversely, the red curve on the right panel meanders through more convoluted paths, yielding a larger $l p$⁠. As we know, the right panel exhibits a smaller thermal conductance compared to the left.

To quantify this observation, we computed the average shortest path distance for each pair of source and sink nodes along both the bottom and upper boundaries. We define the average shortest path distance, denoted as $L ¯$⁠, as $L ¯= 1 N ∑ p = 1 N l p$⁠. Here, $p$ represents each pair of heat source and sink nodes on the opposite boundaries, and $l p$ is the shortest distance between the respective pairs. Given that we are dealing with networks of different sizes characterized by varying boundary lengths, denoted as $N×N$⁠, the average shortest path distance $L ¯$ is expected to increase with the network boundary length $N$⁠. To account for the impact of network size, the average shortest path distance $L ¯$ for each sized network is further normalized by dividing it by the network size $N$⁠, resulting in $L n o r m= L ¯/N$⁠. In our exploration, we successfully tuned amorphous networks with dimensions ranging from $16×16$ to $32×32$ to exhibit both increased and decreased thermal conductance. As shown in Fig. 3(c), the scatter log–log plot reported the logarithmic values of $C$ and $L n o r m$ for different sized amorphous networks. It is noted that these scatters merge together to form a line, indicating the $C$ and $L n o r m$ following the power law. Furthermore, these networks can be accurately fitted with a red linear regression line, expressing the power law relationship between $C$ and $L n o r m$ as approximately $C= L n o r m α$⁠, with $α=−21.46$⁠. The negative power $α$ provides additional evidence that an increase in the path distance through which heat energy travels along the network leads to a reduction in the overall thermal conductance. Therefore, we have proved that the entire thermal conductance of the amorphous network is inversely related to the average shortest path distance between high-temperature and sink nodes. This observation underscores the significance of average path distance as a crucial metric for comprehending variations in heat transfer properties within our amorphous network systems.

We have further uncovered that thermal conductance $C$ adheres to a power law concerning the normalized average shortest distance $L n o r m$⁠, as depicted in the log–log plot in Fig. 3(c). Notably, diverse-sized amorphous networks exhibit a convergence, indicating a common underlying physics. It is noteworthy that the thermal conductance of the amorphous network is highly sensitive to changes in the average shortest distance from high-temperature to sink nodes. Even a slight variation in $L n o r m$⁠, such as a decrease from approximately $1.07$ to $1.02$⁠, constituting only $0.05$ in changes and roughly $5%$ change ratio, can result in tuning $C$ from around $1$ to $3.5$⁠. Therefore, it can be anticipated that finding a method to further decrease $L n o r m$ significantly would lead to a more substantial increase in $C$⁠.

The variation in the shortest path distance can be interpreted as the curvature degree of the path. The presence of more zigzags in the shortest path indicates a more winding trajectory, resulting in a longer path length. This concept is visually represented in Fig. 3(d). For each shortest path, the curvature degree can be quantified by calculating the summation of the angle $θ$ (representing the angle between the bond orientation and the y-axis). To articulate this, we express the summation angle for each bond as $∑ θ b$⁠, where $b$ ranges from 1 to the total number of bonds along the shortest path. Subsequently, we derive the average value of each angle summation $∑ p = 1 N(∑ θ b)$ and normalize it based on the largest angle summation $∑ θ b$ within each network, denoted as $θ n o r m$⁠. A higher $θ n o r m$ indicates a tendency for bonds to align perpendicularly to the heat transport direction, resulting in a decrease in the thermal conductance $C$⁠. Conversely, a smaller $θ n o r m$ implies a more direct alignment of bonds with the heat transport direction, leading to an increase in the thermal conductance $C$ for the amorphous networks.

The planar network graphs can be further transformed into tangible physical entities, specifically metallic networks, as depicted in Fig. 4. Copper was chosen as the material for the entire network, with each bond having a uniform thickness of 0.1 cm. This conversion process resulted in two-dimensional metallic networks, showcased in the first row of Fig. 4. Subsequently, the planar graphs were extended by 1 cm along the $z$ direction to create three-dimensional metallic networks, depicted in the second row of Fig. 4. The effective thermal conductivity (⁠ $κ e f f$⁠) of each network is recorded atop each panel. Due to the presence of multiple voids within the material, the effective thermal conductance values are inherently smaller than the thermal conductivity of a pure copper plate. Notably, the ratio of $κ e f f$ between the optimized networks (both increased and decreased) and the original network, for both two-dimensional and three-dimensional calculations, is approximately 1.30. Similarly, the ratio of $κ e f f$ between the original and decreased networks, for both two-dimensional and three-dimensional calculations, also hovers around 1.30. Moreover, the theoretically calculated ratios of thermal conductance further corroborate these findings, with $C i n c r e a s e/ C o r i g i n a l=1.67$ and $C o r i g i n a l/ C d e c r e a s e=1.63$⁠. These results validate the optimization of network structures, demonstrating that the metallic networks exhibit enhanced or reduced $κ e f f$⁠, corresponding to increased or decreased thermal conductance values. This validation underscores the efficacy of our optimization method in designing metallic network systems.

By employing the SA algorithm, we have successfully customized the thermal conductance of amorphous networks, achieving both increased and decreased values. Throughout the tuning process, the algorithm generated a diverse set of network structures, forming the dataset instrumental in exploring the physical mechanisms governing the heat transfer properties of amorphous networks. Specifically, we have unveiled the crucial role of network bonds as pathways guiding heat flow within the network. This underscores the role of machine learning in the study of heat transport in complex networks. More importantly, this process can be reversed so that the revealed physics mechanism can reciprocally enhance the performance of machine learning model. Recent studies have exemplified this philosophy by employing the Laplacian matrix or Dynamical matrix as input to train the convolutional neural network, which has greatly improved the training and prediction efficiencies.^44,45

With the rapid advancements in nanotechnology, the fabrication of nanowires/nanotubes in the laboratory has become more accessible.^46–50 Metallic seamless nanowire networks, in particular, have garnered increasing interest due to their fast lithographic techniques for fabrication and unique advantages, including being interwire-contact-junction free.^51,52 While their thermal performance has been extensively evaluated based on factors such as nanowires’ aspect ratio,^53,54 network area fraction,⁵⁵ and network density,^56,57 few studies have delved into the impact of geometrical topology on thermal transport properties due to the lack of efficient design protocols. Our results offer insights into the preparation principles of seamless nanowires concerning the topology of the entire network structure. As a result, our discovery linking amorphous network topology to thermal transport behavior may contribute valuable guidance to the fabrication of nanowires/nanotubes with desired thermal properties.

In summary, the presented research unveils a comprehensive methodology for tuning the thermal conductance of amorphous networks, showcasing the potential for broader applications beyond thermal properties. The devised tuning approach not only successfully enhances but also diminishes the thermal conductance of networks, providing a versatile tool for tailoring various characteristics within amorphous systems. Crucially, the tuned networks exhibit distinctive bond orientation tendencies aligned with the predefined thermal transport direction. These tendencies indicate that the thermal conductance of the amorphous network is determined by the average shortest path length between the hottest and the coldest nodes. The observed bond orientation tendencies also provide valuable insights for the experimental creation of nanotubes and nanowires, offering a pathway for the deliberate construction of nanoscale networks with predetermined thermal transfer properties.

This work was supported in part by the National Natural Science Foundation of China (NNSFC) (Grant Nos. 12205138 and 52250191) and Shenzhen Science and Technology Innovation Committee (SZSTI) (Grant/Award No. JCYJ20220530113206015).

The authors have no conflicts to disclose.

Changliang Zhu: Data curation (lead); Formal analysis (equal); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). Tianlin Luo: Formal analysis (equal); Methodology (equal); Writing – review & editing (equal). Baowen Li: Funding acquisition (equal); Supervision (supporting); Writing – review & editing (equal). Xiangying Shen: Conceptualization (lead); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Guimei Zhu: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.