Composable neural emulators accelerate thermoelectric generator design.

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Learn more: PMC Disclaimer | PMC Copyright Notice Nature . 2026 Apr 15;652(8110):643–649. doi: 10.1038/s41586-026-10223-1 Search in PMC Search in PubMed View in NLM Catalog Add to search Composable neural emulators accelerate thermoelectric generator design Airan Li Airan Li 1 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan Find articles by Airan Li 1, # , Xinzhi Wu Xinzhi Wu 1 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan Find articles by Xinzhi Wu 1, # , Longquan Wang Longquan Wang 1 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan Find articles by Longquan Wang 1, # , Gang Wu Gang Wu 1 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan Find articles by Gang Wu 1 , Jiankang Li Jiankang Li 1 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan 2 Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Japan Find articles by Jiankang Li 1, 2 , Zhao Hu Zhao Hu 1 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan 2 Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Japan Find articles by Zhao Hu 1, 2 , Xinyuan Wang Xinyuan Wang 1 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan 2 Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Japan Find articles by Xinyuan Wang 1, 2 , Takao Mori Takao Mori 1 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan 2 Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Japan Find articles by Takao Mori 1, 2, ✉ Author information Article notes Copyright and License information 1 Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan 2 Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Japan ✉ Corresponding author. # Contributed equally. Received 2025 Sep 28; Accepted 2026 Feb 2; Issue date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13083250  PMID: 41986625 Abstract Designing high-performance thermoelectric (TE) devices is challenging because it requires not only advanced materials but also optimal configurations, which are critical for maximizing device performance but remain time-consuming and resource-intensive to identify 1 – 5 . Here we develop TEGNet, a neural network emulator that predicts TE generator performance with greater than 99% accuracy while using only 0.01% of the computational time required by commercial finite-element solvers. TEGNet exhibits strong architectural generality across various material systems and allows flexible combinations of material-specific emulators, unlocking rapid and accurate exploration of diverse device architectures. Using TEGNet, we experimentally optimize MgAgSb/Bi 0.4 Sb 1.6 Te 3 segmented and Mg 3 Bi 1.4 Sb 0.6 –MgAgSb n–p paired TE generators, achieving conversion efficiencies of 9.3% and 8.7%, respectively, ranking competitively high among those previously reported 6 – 10 . This work demonstrates the power of artificial intelligence (AI) in TE generator design, inspiring further research on AI for thermoelectrics. Subject terms: Thermoelectric devices and materials, Thermoelectrics A composable neural network emulator is described for speeding up thermoelectric generator design, demonstrating the ability to predict generator performance with >99% accuracy while taking only 0.01% of the time compared with commercial finite-element solvers. Main TE technology enables direct solid-state conversion of heat into electricity and offers a promising solution for waste heat recovery across a wide temperature range and diverse application scenarios 11 – 15 . A typical TE generator consists of several TE legs electrically connected in series and thermally in parallel. Each leg comprises a TE material sandwiched between interface materials 9 , 16 , 17 . The performance of TE materials is characterized by the dimensionless figure of merit, zT = S 2 σT / κ , in which S , σ and κ represent the Seebeck coefficient, electrical conductivity and thermal conductivity of the material, respectively, and T is the absolute temperature. Intensive efforts over recent decades have led to a wide range of high-performance TE materials, with reported zT values exceeding 1 and even approaching 3 (refs. 18 – 24 ). However, achieving high-performance TE generators requires more than the selection of superior materials. It necessitates a comprehensive approach that simultaneously addresses material compatibility, geometric tuning and architectural optimization 2 – 5 . The device efficiency is influenced by the TE compatibility factor, defined as s = ((1 + zT ) 1/2 − 1)/( ST ), in which matching the operating relative current density to the local s is important to minimize irreversible energy losses and maximize the zT of the materials 1 . Device geometry, including the leg length H and cross-sectional area A , also plays a critical role, as it directly governs the coupled transport of electrical current density and the heat flux: electrical resistance R scales with H / A , whereas the thermal conductance K scales with A / H . In practice, semi-analytical approaches based on a compatibility-factor-matching strategy are often combined with finite-element simulations, enabling consideration of temperature-dependent σ ( T ), S ( T ), κ ( T ) and various parasitic losses for full-parameter optimization and minimized energy-loss integration 3 . Nevertheless, substantial optimization space remains: maximizing both conversion efficiency and power density requires carefully balancing the power factor PF = S 2 σ , thermal transport and geometric parameters, a task that cannot be achieved by maximizing zT alone 25 . Therefore, delicate TE generator design is critical to fully exploit material potential. However, this process remains complex and challenging, as it requires several optimization strategies and involves time-consuming and resource-intensive computation and experimentation. Traditionally, finite-element simulations are reliable approaches, enabling designers to virtually combine different TE materials and perform systematic scans of leg dimensions under various boundary conditions to determine the optimal configuration before fabrication 26 – 29 . However, simulating TE generator performance requires repeatedly setting up models and solving coupled thermo-electrical partial differential equations (PDEs), which becomes time-consuming and resource-intensive when material choices, geometries, boundary conditions and device architectures change frequently. Recent advances in AI and machine learning have enabled data-driven acceleration of complex design problems. Using neural networks, one of the most powerful and widely used approaches in AI and machine learning, as surrogates for solving PDEs is particularly promising. Once trained, they enable ultrafast forward predictions with orders-of-magnitude acceleration. In principle, neural networks can be applied to TE generator design by taking device geometry as input and predicting device performance as output, thereby enabling rapid and accurate optimization of TE devices 30 , 31 . However, given the vast diversity of high-performance TE materials, varying transport properties and different types of device architecture, how to integrate neural networks into TE generator design to efficiently navigate the broad design space remains challenging, despite its considerable potential. In this work, we develop a composable neural network emulator, called TEGNet, as a fast and accurate surrogate for finite-element simulations in TE generator design. TEGNet enables efficient exploration of diverse device architectures, including segmented legs and n–p pairs, by flexibly combining material-specific emulators. Guided by TEGNet, we further demonstrate experimental optimization of representative TE generators, validating its effectiveness for device-level design. TEGNet architecture and model training Conventional TE generator design relies on repeated finite-element simulations (for example, COMSOL, ANSYS). As shown in Fig. 1a , dimensions (Dims) and boundary conditions (BCs) are input into solvers to compute coupled thermo-electrical PDEs, yielding spatial voltage V ( r ) and temperature T ( r ), from which the output voltage V 0 and cold-side heat flow Q 0 can be extracted. However, solving these PDEs is highly time-consuming when Dims and BCs vary, a common requirement in TE generator design. Therefore, developing a surrogate model to replace the PDE-solving process can accelerate simulations and, consequently, device design. Neural networks, with strong nonlinear mapping capability, are especially suitable for thermo-electrical coupling. Notably, once trained, these models can produce outputs ultrafast. Physics-informed neural networks have been used to solve PDEs and extract field-dependent properties, but they require carefully designed loss functions for stability and accuracy 32 . By contrast, TE generator design is simpler, as the focus is mainly on overall output performance rather than detailed field distributions, which allows simpler models and efficient training. On the basis of this consideration, we propose TEGNet, a streamlined neural network (Fig. 1b ) with Dims and BCs as input features and two outputs ( V 0 and Q 0 ). Notably, the outputs contain only the quantities of interest, omitting internal voltage and temperature distributions. Fig. 1. TEGNet architecture and model training. Open in a new tab a , Workflow of conventional simulation and the concept of a surrogate model. b , Schematic diagram of TEGNet with simplified neurons. c , Epoch-dependent mean training and validation loss over five independent data splits when the sample size is 1,200. d , e , Root-mean-square error (RMSE; d ) and mean absolute error (MAE; e ) of predicted V 0 and Q 0 on the test set for different sample sizes. f , g , Scatter plots of predicted versus true V 0 ( f ) and predicted versus true Q 0 ( g ) on the test set when the sample size is 1,200. To train TEGNet, we select MgAgSb, a promising TE material to replace Bi 2 Te 3 for near-room-temperature power generation, as a case study. Training data are generated by COMSOL simulations, which are computationally expensive initially but greatly reduce cost once TEGNet is established. Because sample size strongly influences accuracy and generalization, we examine its effect to determine an optimal balance between performance and cost. Data were split into training (80%), validation (10%) and test sets (10%) first to validate the parameters of TEGNet. As shown in Fig. 1c and Supplementary Fig. 1 , both training and validation losses decrease and saturate with increasing epochs in all sample sizes, confirming suitable model parameters. Model performance on the test set (Fig. 1d,e ) shows consistently low root-mean-square error and mean absolute error for V 0 and Q 0 , along with a decreasing trend in these errors as sample size increases, indicating improved generalization. This trend is expected because more data help the model learn the underlying patterns more reliably. Considering both data-generation cost and model accuracy, a sample size of 1,200 is determined to be optimal. With this sample size, the coefficient of determination R 2 reaches 0.9997 and 0.9998 for V 0 and Q 0 predictions, respectively, on the unseen test set (Fig. 1f,g ), demonstrating both the sufficiency of this sample size compared with others (Supplementary Fig. 2 ) and the robustness of the TEGNet architecture. Accurate and rapid simulations using TEGNet With the validated TEGNet, we next evaluate its capability in TE generator simulations. Using the MgAgSb-data-trained TEGNet, we compare predictions with COMSOL by varying T h and I , following a typical TE generator measurement procedure. As shown in Fig. 2a , the I -dependent V 0 and Q 0 predicted by TEGNet closely reproduce the expected TE generator behaviour and perfectly match COMSOL results. Scatter plots (Fig. 2b ) further show the excellent agreement, with R 2 = 0.9999 for V 0 and R 2 = 0.9997 for Q 0 . From the TEGNet predictions, output power P and conversion efficiency η can be derived (Fig. 2c ). Similarly, TEGNet predictions show excellent agreement with COMSOL, achieving R 2 = 0.9997 for P and R 2 = 0.9993 for η (Fig. 2d ). Notably, TEGNet focuses on intrinsic outputs ( V 0 , Q 0 ), from which P and η are derived using P = I × V 0 and η = P /( P + Q 0 ). The maximum power P max and maximum conversion efficiency η max can then be extracted from I – P and I – η curves under various T h (Supplementary Fig. 3 ). This intrinsic prediction mechanism not only improves interpretability but also lays the foundation for the composable capability of TEGNet, enabling application across diverse device architectures, a feature we demonstrate in the following sections. Although TEGNet is trained within a fixed temperature range, it also exhibits reasonable extrapolation beyond this range, as shown in Supplementary Fig. 4 . Moreover, TEGNet exhibits a unique ability to rapidly predict performance under BCs beyond T h and T c . For example, TE generators often operate under constant heat flux. With TEGNet, we can rapidly search for combinations of T h , T c and I that satisfy a constant heat-flux requirement without solving any PDEs. As shown in Extended Data Fig. 1 , the predictions of TEGNet under constant heat flux agree closely with COMSOL results obtained by directly imposing heat-flux inputs and solving the full PDEs. Fig. 2. Accurate and rapid TEGNet simulation. Open in a new tab a , I -dependent V 0 and Q 0 predicted by TEGNet and obtained from COMSOL under different T h for MgAgSb. b , Scatter plots of V 0 and Q 0 predicted by TEGNet versus those obtained by COMSOL. c , I -dependent P and η calculated from I – V 0 and I – Q 0 predicted by TEGNet and obtained from COMSOL under different T h for MgAgSb. d , Scatter plots of P and η from TEGNet versus COMSOL results. e , f , Scatter plots of P max ( e ) and η max ( f ) from TEGNet versus COMSOL for various TE materials under different T h . g , Computational time required to obtain P max and η max under different T h using COMSOL and TEGNet for different TE materials. Extended Data Fig. 1. Open in a new tab I -dependent V 0 and Q 0 predicted by TEGNet and obtained from COMSOL under constant heat-flux conditions for MgAgSb, along with scatter plots comparing TEGNet predictions to COMSOL results for both V 0 and Q 0 . To further test the generality of TEGNet, we trained TEGNet on a broad set of TE materials beyond MgAgSb, including Bi 0.4 Sb 1.6 Te 3 , Mg 3 Bi 1.4 Sb 0.6 , Mg 3 Sb 1.5 Bi 0.5 , NbFeSb, GeTe, PbTe, CoSb 3 , AgCu(Te,S), Zintl phases, SnSe and SnS (refs. 33 – 42 ), which cover broad TE performance regimes and operating temperature ranges (Supplementary Fig. 5 ). As shown in Supplementary Figs. 6 – 16 , the I -dependent V 0 , Q 0 , P and η obtained from TEGNet align well with COMSOL and extracted P max and η max (Supplementary Figs. 17 and 18 ) also show strong agreement. Figure 2e,f summarizes the scatter plots of extracted P max and η max , which yield R 2 of 0.9999 and 0.9996, respectively. These results demonstrate the excellent architectural generality of TEGNet across a wide range of TE materials. Notably, as introduced earlier, the key advantage of TEGNet lies in its speed. As shown in Fig. 2g , COMSOL requires about 2,237.42 s per material’s simulation, whereas TEGNet predicts the same outputs in only 0.25 s (just 0.01% of the time). Although initial data generation (about 4 h for 1,200 samples) and training (around 79 s) are required, these are one-time costs and are modest compared with the overall development cycle of TE materials and devices. Importantly, once trained, TEGNet enables ultrafast predictions and offers high reusability and flexibility for repeated and diverse TE generator designs, as demonstrated in the following sections. Notably, for materials with TE properties similar to those above, TEGNet can also provide a reasonable rough estimate, as illustrated by the MgAgSb example, whereas for entirely new materials or cases requiring precise predictions, TEGNet models can also be readily constructed by following the same workflow presented in this study, as demonstrated above (Supplementary Fig. 19 ). TEGNet application for segmented TE generator design Unlike previous models 30 , 31 , the key advantage of TEGNet lies in its composability, enabled by accurate predictions of intrinsic V 0 and Q 0 under varied Dims and BCs. The operating current I in the BCs serves as the physical foundation for this composability. A TE generator is an electronic system in which the same I flows through all segments during operation. This constant- I rule allows individual unit-level TEGNet models to be directly combined to construct system-level simulations. As illustrated in Fig. 3a , when two materials are connected in series to form segmented generators, the overall voltage equals V 0_1 + V 0_2 , whereas the overall heat flow equals Q 0_2 at the cold side. The interface temperature T i between segments, initially unknown, can be solved by the energy conservation rule ( Q 0_1 = Q 0_2 + I × V ). It should be emphasized that this framework is not limited to two-segment TE generators but can be readily extended to three or more segments, offering a flexible and efficient way to design any high-performance segmented TE generators. Fig. 3. Segmented TE generator design framework and its application. Open in a new tab a , Framework for a typical segmented TE generator design. b , c , T -dependent zT ( b ) and PF ( c ) of MgAgSb and Bi 0.4 Sb 1.6 Te 3 . d , e , The ratio of MgAgSb length to total length ( c MgAgSb / c )-dependent η max ( d ) and P max ( e ) under different total lengths of the segmented TE generator predicted by TEGNet. f , Scatter plot of normalized P max and η max from TEGNet versus COMSOL. g , Computational time for obtaining P max and η max by TEGNet and COMSOL. h , Experimental probe distance-dependent resistance of segmented MgAgSb/Bi 0.4 Sb 1.6 Te 3 . i , Experimental power density of Bi 0.4 Sb 1.6 Te 3 , segmented MgAgSb/Bi 0.4 Sb 1.6 Te 3 and MgAgSb under different total lengths. j , Experimental I -dependent η of segmented MgAgSb/Bi 0.4 Sb 1.6 Te 3 under different T h . To better demonstrate the capability of TEGNet in assisting segmented TE generator design, we directly simulated Bi 0.4 Sb 1.6 Te 3 /MgAgSb segmented generators. Both materials exhibit high performance near room temperature but distinct peak zT and PF at different temperatures (Fig. 3b,c ), which makes the optimal design non-intuitive without simulation. With TEGNet, we efficiently scanned different length ratios and extracted η max and P max (Fig. 3d,e ). Results from the composed TEGNet closely match COMSOL ( R 2 = 0.9990; Fig. 3f ). The detailed length-dependent P max and η max obtained from COMSOL are provided in Supplementary Fig. 20 . Notably, despite the extra T i -solving step, computation remains fast in the composed TEGNet, requiring only 74 s to identify the optimal configuration versus a much longer time in COMSOL (Fig. 3g ). For experimental validation, we fabricated Bi 0.4 Sb 1.6 Te 3 /MgAgSb segmented generators, selecting a 0.5 length ratio for MgAgSb owing to its superior simulated output (Fig. 3d,e ). Sb was used as the interface layer 43 and the measurement set-up for contact resistivity and power output is shown in Supplementary Fig. 21 . The measured contact resistivities are 4.0 μΩ cm 2 for Sb/MgAgSb and 4.9 μΩ cm 2 for Bi 0.4 Sb 1.6 Te 3 /In–Ga/Sb (Fig. 3h ), both sufficiently low for practical application. As shown in Fig. 3i , the segmented generator exhibits higher power density than single legs, consistent with simulations. Moreover, it achieves conversion efficiencies of 9.3% at 593 K, ranking among the best reported 6 , 7 (Fig. 3j ). Moreover, beyond assisting segmented TE generator design, the composability of TEGNet also enables the inclusion of electrical contact resistivity ρ c for practical device considerations. As shown in Extended Data Fig. 2 , predictions from TEGNet incorporating ρ c remain in excellent agreement with COMSOL results. As well as ρ c , TEGNet can also account for thermal contact resistivity r c (Extended Data Fig. 3 ). Notably, when incorporating these parasitic effects, whether ρ c or r c , there is no need to retrain TEGNet or solve PDEs, while still achieving COMSOL-level accuracy, further enhancing its capabilities and underscoring its strong potential for real-world applications. Extended Data Fig. 2. Open in a new tab I -dependent V 0 and Q 0 predicted by TEGNet and obtained from COMSOL with incorporation of electrical contact resistivity for MgAgSb and scatter plots of V 0 and Q 0 predicted by TEGNet versus those obtained by COMSOL. Extended Data Fig. 3. Open in a new tab I -dependent V 0 and Q 0 predicted by TEGNet and obtained from COMSOL with incorporation of thermal contact resistivity for MgAgSb and scatter plots of V 0 and Q 0 predicted by TEGNet versus those obtained by COMSOL. TEGNet application for n–p paired TE generators Beyond segmented generators, TEGNet can also be composed to design n–p paired TE generators. As shown in Fig. 4a , the overall V 0 and Q 0 can be obtained by summing the contributions from individual TEGNet models, as the TE legs are connected electrically in series and thermally in parallel. By varying the dimensions of n-type and p-type legs and adjusting boundary conditions, we can efficiently explore optimal configurations that balance internal resistance and heat flow to improve η or P . To demonstrate this, we paired Mg 3 Bi 1.4 Sb 0.6 (n-type) with MgAgSb (p-type). It should be noted that both TEGNet models had been trained previously, further highlighting their reusability across different design tasks. Moreover, scaling an n–p pair to several pairs requires no further computational cost, as the device-level performance can be obtained simply by summing the corresponding TEGNet outputs (Supplementary Fig. 22 ). Using the composed framework, we rapidly computed the η max and P max as functions of the area ratio ( A p / A n ) and normalized height H /( A p + A n ) under T c = 293 K and T h = 573 K (Fig. 4b,c ), in which A p , A n and H represent the cross-sectional area of the p-type leg, the n-type leg and their common height, respectively. Figure 4d,e presents these results as 2D contour maps, providing a more intuitive view of the design trends. Notably, TEGNet predictions show excellent agreement with COMSOL results for H /( A p + A n ) = 0.28 or A p / A n = 1.28 (Fig. 4f ); while COMSOL required more than 5,800 s to simulate sparse data points, TEGNet generated the full maps in only 9 s. The results from Fig. 4d,e indicate that higher η max can be achieved when A p / A n is around 1.3, whereas increasing H /( A p + A n ) leads to a reduction in P max . Moreover, varying A p + A n and including electrical contact resistivity do not obviously affect the optimal A p / A n of 1.3 for achieving high η max (Supplementary Figs. 23 and 24 ). Fig. 4. The n–p paired TE generator design framework and its application. Open in a new tab a , Framework for a typical n–p paired TE generator design. b , c , 3D plots of η max ( b ) and P max ( c ) as functions of A p / A n and H /( A p + A n ) for the Mg 3 Bi 1.4 Sb 0.6 –MgAgSb n–p paired TE generator predicted by TEGNet. d , e , Corresponding 2D contour plots of η max ( d ) and P max ( e ). f , Scatter plot of normalized P max and η max from TEGNet versus COMSOL. g , Optical image of the fabricated Mg 3 Bi 1.4 Sb 0.6 –MgAgSb two n–p paired TE generator and its measurement set-up. h , i , I -dependent V 0 and P ( h ) and Q 0 and η ( i ) for the fabricated Mg 3 Bi 1.4 Sb 0.6 –MgAgSb two n–p paired TE generator under different T h . j , T -dependent η max for the fabricated Mg 3 Bi 1.4 Sb 0.6 –MgAgSb two n–p paired TE generator and comparison with other TE generators based on different TE material systems 19 , 29 , 44 , 45 . k , 3D and 2D contour plots of η max as functions of A p / A n and H /( A p + A n ) for the Mg 3 Sb 1.5 Bi 0.5 –SnS n–p paired TE generator predicted by TEGNet. l , 2D contour plots of η max of n-type segmented Mg 3 Sb 1.5 Bi 0.5 /Mg 3 Bi 1.4 Sb 0.6 –p-type GeTe paired TE generator as functions of leg length ratio and geometry predicted by TEGNet. Guided by these simulations, we fabricated a two-pair TE generator with p-type legs of 3.3 × 3.3 × 6.6 mm 3 and n-type legs of 2.9 × 2.9 × 6.6 mm 3 to potentially improve η max . The optical image of the TE generator and the measurement set-up are shown in Fig. 4g . Measurements (Fig. 4h,i ) show that η max reaches 8.7% when T h = 593 K, which surpasses our previous values 7 , 33 , 43 and ranks competitively high among those reported in the literature 8 – 10 . Notably, the Mg 3 Bi 1.4 Sb 0.6 –MgAgSb paired TE generator also outperforms other material combinations in the near-room-temperature range 19 , 29 , 44 , 45 (Fig. 4j ), highlighting its strong potential for harvesting low-grade heat. As noted above, the composability of TEGNet enables the integration of different material-specific emulators and supports diverse device architectures, including three-segment designs and segmented n–p paired generators (Supplementary Fig. 25 ). To further showcase the flexibility of TEGNet, we also simulated an n–p pair using Mg 3 Sb 1.5 Bi 0.5 and SnS (Fig. 4k ), and a segmented n-type Mg 3 Sb 1.5 Bi 0.5 /Mg 3 Bi 1.4 Sb 0.6 combined with a p-type GeTe (Fig. 4l ). Unlike conventional COMSOL modelling, which requires building new 3D models and solving PDEs, these simulations were completed within about 30 s by simply composing trained TEGNet models. Notably, on the basis of the results, the optimal configurations of TE generator often deviate from the conventional set-up with equal p-type and n-type cross-sectional areas, further highlighting the importance of careful design for maximizing generator performance. Moreover, the composable nature of TEGNet enables simultaneous optimization of both geometrical parameters and material choices. By evaluating all possible combinations of different n-type and p-type TEGNet models, their best performance can be rapidly assessed (Supplementary Fig. 26 ). With the incorporation of more material-specific TEGNet models, it will become feasible to screen a large number of n–p combinations. Conclusion This work demonstrates, both theoretically and experimentally, that neural-network-based emulators can serve as a practical and scalable alternative to conventional finite-element workflows for TE generator design. By enabling rapid exploration of diverse device architectures while maintaining high predictive fidelity, including the incorporation of parasitic effects, TEGNet provides a generalizable methodology that accelerates the translation from materials discovery to device implementation, highlighting the potential of AI-driven approaches in TE research. However, despite its strong capabilities, TEGNet is not intended to address all aspects of TE generator design or to uncover new physical rules at this stage. Instead, it focuses on rapid exploration of the broad device design space, rather than detailed field-level, nonlinear, transient or stress-related effects, which may be incorporated in future extensions. By combining physical principles with data-driven modelling, TEGNet addresses a critical and previously unmet methodological gap in AI-assisted TE device design. The demonstrated application of AI at the device level is expected to inspire deeper integration of AI into TE research, complementing material-level efforts and advancing innovation in AI for thermoelectrics and energy applications. Methods TEGNet architecture The TEGNet architecture is a streamlined, fully connected, multilayer perceptron, consisting of one input layer, three hidden layers and one output layer. The input layer contains six neurons representing the dimensions of a TE leg (width a , depth b and length c ) and the boundary conditions (hot-side temperature T h , cold-side temperature T c and applied current I ). The output layer has two neurons corresponding to the voltage V 0 and heat flow Q 0 at the cold side. Each hidden layer contains 128 neurons, with SiLU as the activation function. The loss function is defined as the mean squared error between the predicted values and the data generated from COMSOL Multiphysics (abbreviated as COMSOL in the main text and the following), allowing for backpropagation during training. All algorithms were implemented in Python (version 3.10) using the PyTorch module. Data generation The training data for TEGNet is generated using COMSOL owing to its high accuracy in solving coupled thermo-electrical PDEs. All simulations used a ‘normal’ mesh size and a steady-state solver. Each data point contains eight columns: the TE leg dimensions ( a , b and c ), boundary conditions ( T h , T c and I ) and the corresponding V 0 and Q 0 . The dimensions a , b and c range from 1 to 12 mm. Temperature ranges for T h and T c were selected according to the intrinsic operating limits of each material to avoid unreal TE properties arising from material degradation or phase changes. An extra 20 K margin was included to ensure full coverage of the TE property data and allow reasonable extrapolation. For example, NbFeSb operates up to about 1,203 K, so T h and T c were sampled in the range 283–1,223 K, with T h > T c . For MgAgSb, which operates up to about 573 K, the corresponding range was 283–593 K. The Sobol sequence is used to sample a , b , c , T c and T h , for which T h should exceed T c . The maximum current I is dynamically determined on the basis of a , b , c , T c , T h and the TE properties of the material to avoid non-convergent COMSOL cases and ensure operation within the power-generation regime. The current I is then sampled uniformly from 0 to the maximum value. These considerations effectively eliminate unphysical conditions and yield a high-quality dataset requiring minimal preprocessing. These combined input parameters ( a , b , c , T c , T h , I ) are fed into COMSOL and V 0 and Q 0 are extracted from the solved spatial distributions of voltage and temperature. Because n-type and p-type materials produce voltages of opposite sign under identical temperature gradients, all COMSOL-generated V 0 values were converted to positive. Sample sizes of 300, 600, 1,200, 1,800 and 2,400 were tested for MgAgSb and an optimal sample size of 1,200 was chosen on the basis of a balance between accuracy and computational cost for all TE materials. TEGNet training For TEGNet training, the input data are fed into the network and the weights and biases of each neuron are updated by means of backpropagation to minimize the loss function. The optimizer is Adam and a CosineAnnealingLR scheduler is used with an initial learning rate of 1 × 10 −3 . The total number of epochs is set to 6,000. The TEGNet parameters were initially evaluated using MgAgSb data, which were split into a training set (80%), a validation set (10%) and a test set (10%). The validation set serves as an indicator for overfitting or underfitting during training. The model performance was evaluated after confirming the suitability of the network parameters. The data were divided into a training set (80%) and a test set (20%) to fully use the available data. To reduce potential bias from data splitting, five independent random splits were performed for both parameter evaluation and model performance assessment and the results were averaged. The final model for each material was trained on the entire dataset to maximize data use. Through training, the temperature-dependent TE properties are effectively learned and embedded within the network weights for each material. All training was conducted on a personal computer with a 12-core CPU and 16 GB RAM. Materials synthesis The MgAgSb used in this work contains 0.625 wt% C 18 H 36 O 2 and Mg 3 Bi 1.4 Sb 0.6 has the composition Mg 3.2 In 0.02 Sb 0.595 Bi 1.4 Te 0.005 , whereas Bi 0.4 Sb 1.6 Te 3 is used as is. For simplicity, MgAgSb + 0.625 wt% C 18 H 36 O 2 and Mg 3.2 In 0.02 Sb 0.595 Bi 1.4 Te 0.005 are abbreviated as MgAgSb and Mg 3 Bi 1.4 Sb 0.6 , respectively, throughout the main text and the following. MgAgSb and Mg 3 Bi 1.4 Sb 0.6 were prepared by ball milling (SPEX 8000D) in an Ar atmosphere for 5 h, whereas Bi 0.4 Sb 1.6 Te 3 was prepared by melting at 1,273 K in a quartz tube for 12 h, followed by ball milling of the resulting ingot for 1 h. The raw materials used are high-purity Mg turnings (99.95%), Ag powders (99.99%), Sb shots (99.999%), Bi shots (99.999%), Te shots (99.999%) and In powders (99.99%). TE generator fabrication For the MgAgSb/Bi 0.4 Sb 1.6 Te 3 segmented TE generator, the MgAgSb part was fabricated with Sb powders as interface materials, whereas the Bi 0.4 Sb 1.6 Te 3 part was fabricated without interface materials. Both components were spark plasma sintered using SPS-322LX (Dr. Sinter) at 573 and 693 K for 5 and 10 min under 60 MPa, respectively. The Sb/MgAgSb/Sb piece was cut parallel to the pressing direction, whereas Bi 0.4 Sb 1.6 Te 3 was cut perpendicular to the pressing direction. Segmented TE generators with different dimensions were assembled by connecting Sb/MgAgSb/Sb and Bi 0.4 Sb 1.6 Te 3 using liquid Ga–In to form Sb/MgAgSb/Sb/Ga–In/Bi 0.4 Sb 1.6 Te 3 , which is abbreviated as MgAgSb/Bi 0.4 Sb 1.6 Te 3 in the main text for simplicity. The contact resistance of the Sb/MgAgSb/Sb/Ga–In/Bi 0.4 Sb 1.6 Te 3 interface was measured using a resistance distribution instrument (S1331, Mottainai Energy). Notably, even without interface materials, Bi 0.4 Sb 1.6 Te 3 in contact with Sb through liquid Ga–In exhibited negligible contact resistance. For the Mg 3 Bi 1.4 Sb 0.6 –MgAgSb n–p paired TE generator, the Mg 3 Bi 1.4 Sb 0.6 part was fabricated with stainless-steel powders as interface materials and sintered at 973 K under 60 MPa for 10 min (SPS-1080 System, SPS Syntex Inc.). The MgAgSb part was prepared in the same manner as for the segmented TE generator. Two-pair Mg 3 Bi 1.4 Sb 0.6 –MgAgSb n–p paired TE generators were assembled on an AlN ceramic plate with copper electrodes. TE generator measurement The experimental current I -dependent voltage V 0 and heat flow Q 0 of both the MgAgSb/Bi 0.4 Sb 1.6 Te 3 segmented TE generator and the Mg 3 Bi 1.4 Sb 0.6 –MgAgSb n–p paired TE generator were measured using a Mini-PEM system (ADVANCE RIKO). During measurements, the top side was heated to a set temperature to serve as the hot side, whereas the bottom side was maintained at 293 K as the cold side. The hot-side temperature was varied between 373 and 593 K to investigate generator performance under different temperature differences. I was varied to obtain the corresponding V 0 and Q 0 . All measurements were conducted under vacuum conditions. The output power P and conversion efficiency η were calculated as P = I × V 0 and η = P /( P + Q 0 ), respectively. The maximum power P max and maximum conversion efficiency η max for each temperature difference were extracted from the corresponding I – P and I – η curves. Online content Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41586-026-10223-1. Supplementary information Supplementary Information (10.3MB, pdf) Supplementary Figs. 1–26 and Supplementary References. Peer Review File (5.4MB, pdf) Acknowledgements This work was supported by the JST-Mirai Program (JPMJMI19A1). Institutional support from the JSPS WPI Academy Program is also acknowledged. Extended data figures and tables Author contributions A.L. and X. Wu conceived and designed the AI scheme. A.L. conducted the model set-up, training, data generation, materials synthesis (MgAgSb and Bi 0.4 Sb 1.6 Te 3 ), TE generator fabrication and performance measurements, with assistance from X. Wu and G.W. L.W. provided the Mg 3 Bi 1.4 Sb 0.6 materials. A.L. and X. Wu analysed the data, with support from J.L., Z.H. and X. Wang. A.L. drafted the original manuscript and T.M. supervised the entire project. All authors discussed, reviewed and edited the manuscript. Peer review Peer review information Nature thanks Ruomeng Huang, Ady Suwardi and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available. Data availability All data generated or analysed during this study are included in the published article (and its Supplementary Information files). Code availability Models or code generated or used during this study are publicly available at https://github.com/airannims/TEGNet/ . Competing interests The authors declare no competing interests. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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[ Google Scholar ] Associated Data This section collects any data citations, data availability statements, or supplementary materials included in this article. Supplementary Materials Supplementary Information (10.3MB, pdf) Supplementary Figs. 1–26 and Supplementary References. Peer Review File (5.4MB, pdf) Data Availability Statement All data generated or analysed during this study are included in the published article (and its Supplementary Information files). Models or code generated or used during this study are publicly available at https://github.com/airannims/TEGNet/ . 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