Thermal analog computing: Application to matrix-vector multiplication with inverse-designed metastructures

The rising computational demand of modern workloads has renewed interest in energy-efficient paradigms such as neuromorphic and analog computing. A fundamental operation in these systems is matrix-vector multiplication (MVM), ubiquitous in signal processing and machine learning. Here, we demonstrate MVM using inverse-designed metastructures that exploit heat conduction as the signal carrier. The proposed approach is based on a generalization of effective thermal conductivity to systems with multiple input and output ports: The input signal is encoded as a set of applied temperatures, while the output is represented by the power collected at designated terminals. The metastructures are obtained via density-based topology optimization, enabled by a differentiable thermal transport solver and automatic differentiation, achieving an accuracy $>99\%$ in most cases across a pool of matrices with dimensions $2\times2$ and $3\times3$. We apply this methodology -- termed thermal analog computing -- to realize matrices relevant to practical tasks, including the discrete Fourier transform and convolutional filters. These findings open new avenues for analog information processing in thermally active environments, including temperature-gradient sensing in microelectronics and thermal control systems.