To meet such disparate measurement needs, a variety of temperature sensors have been developed. Although these devices vary greatly in their cost, size, weight and complexity, they almost all rely on well-established measurements of transport properties to infer temperature. Legacy technologies like platinum resistance thermometers and negative temperature coefficient thermistors have been relied upon for over a century to provide accurate and reproducible measurements over a broad range of temperature [2-4]. However, these sensors are prone to drift and require frequent re-calibrations to ensure high accuracy in critical use-cases resulting in increase cost of sensor ownership. In recent years, there has been a growing interest in developing alternative sensor technologies that can overcome the limitations of traditional technologies. The past decade has seen a burst of activity in nanophotonics [5], quantum optomechanics [6] and noise thermometry [7]. These technologies leverage telecomm industry's vast economies of scale along with precision measurement expertise developed for frequency metrology to enable fit-for-purpose, cost-effective measurement solutions. Development of an ultra-stable temperature sensor that shows minimal drift over decadal time spans or a field-deployable thermodynamic temperature sensor, likely based on quantum technologies could disrupt the calibration-centered metrology ecosystem of today [4, 5].

Diamond nanoparticles (nanodiamonds) offer an attractive opportunity to achieve high spatial resolution because they can easily be close to the target within a few 10 nm simply by attaching them to its surface [8]. A physical model for such a randomly oriented nanodiamond ensemble (NDE) is available [8], but the complexity of actual experimental conditions still limits the accuracy of deducing magnetic fields. Here, we demonstrate magnetic field imaging with high accuracy of 1.8 µT combining NDE and machine learning without any physical models. We also discover the field direction dependence of the NDE signal, suggesting the potential application for vector magnetometry and improvement of the existing model. Our method further enriches the performance of NDE to achieve the accuracy to visualize mesoscopic current and magnetism in atomic-layer materials [9-13] and to expand the applicability in arbitrarily shaped materials [7], including living organisms [14, 15]. This achievement will bridge machine learning and quantum sensing for accurate measurements. The nitrogen-vacancy (NV) center in diamond [Figure 1(a)] is a point defect where a nitrogen atom replaces a carbon atom in the lattice accompanied by a neighboring vacancy. By measuring its photoluminescence intensity while irradiating the laser and microwaves, NV's electron spin resonance can be detected, which is called optically detected microwave resonance (ODMR) [16]. As the NV's spin level splits against the magnetic field in the direction of the NV symmetry axis (111) due to the Zeeman effect [17], the determination of the ODMR frequency serves as quantum sensing of the field [3]. To obtain a nanoscale spatial resolution, we must attach the NV centers close to the sample within a few 10 nm [18].