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 greater than $99\mathrm{\%}$ in most cases across a pool of matrices with dimensions $2\times 2$ and $3\times 3$. 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 avenues for analog information processing in thermally active environments, including temperature-gradient sensing in microelectronics and thermal control systems.

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• Analog computation
• Heat transfer
• Neuromorphic computing
• Optimization problems
• Physics of computation
• Thermal conductivity
• Metamaterials