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Neural field

In machine learning, a neural field, is a mathematical field that is fully or partially parametrized by a neural network. Initially developed to tackle visual computing tasks, such as rendering or reconstruction, neural fields emerged as a promising strategy to deal with a wider range of problems, including surrogate modelling of partial differential equations, such as in physics-informed neural networks.

Formulation and training
According to the universal approximation theorem, provided adequate learning, sufficient number of hidden units, and the presence of a deterministic relationship between the input and the output, a neural network can approximate any function to any degree of accuracy. Hence, in mathematical terms, given a field \boldsymbol{y} = \Phi(\boldsymbol{x}), with \boldsymbol{x} \in \R^{n} and \boldsymbol{y} \in \R^{m}, a neural field \Psi_{\theta}, with parameters \boldsymbol{\theta}, is such that: In order to overcome this limitation, several strategies have been developed. For example, SIREN uses sinusoidal activations, while the Fourier-features approach embeds the input through sines and cosines. == Conditional neural fields ==
Conditional neural fields
In many real-world cases, however, learning a single field is not enough. For example, when reconstructing 3D vehicle shapes from Lidar data, it is desirable to have a machine learning model that can work with arbitrary shapes (e.g., a car, a bicycle, a truck, etc.). The solution is to include additional parameters, the latent variables (or latent code) \boldsymbol{z} \in \R^d, to vary the field and adapt it to diverse tasks. • Auto-decoding: each training example has its own latent code, jointly trained with the neural field parameters. When the model has to process new examples (i.e., not originally present in the training dataset), a small optimization problem is solved, keeping the network parameters fixed and only learning the new latent variables. Since the latter strategy requires additional optimization steps at inference time, it sacrifices speed, but keeps the overall model smaller. Moreover, despite being simpler to implement, an encoder may harm the generalization capabilities of the model. Specifically, it consists of approximating \Gamma(\boldsymbol{z}) with a neural network \hat\Gamma_{\gamma}(\boldsymbol{z}), where \boldsymbol{\gamma} are the trainable parameters of the hypernetwork. This approach is the most general, as it allows to learn the optimal mapping from latent codes to neural field parameters. However, hypernetworks are associated to larger computational and memory complexity, due to the large number of trainable parameters. Hence, leaner approaches have been developed. For example, in the Feature-wise Linear Modulation (FiLM), the hypernetwork only produces scale and bias coefficients for the neural field layers. Meta-learning Instead of relying on the latent code to adapt the neural field to a specific task, it is also possible to exploit gradient-based meta-learning. In this case, the neural field is seen as the specialization of an underlying meta-neural-field, whose parameters are modified to fit the specific task, through a few steps of gradient descent. An extension of this meta-learning framework is the CAVIA algorithm, that splits the trainable parameters in context-specific and shared groups, improving parallelization and interpretability, while reducing meta-overfitting. This strategy is similar to the auto-decoding conditional neural field, but the training procedure is substantially different. == Applications ==
Applications
Thanks to the possibility of efficiently modelling diverse mathematical fields with neural networks, neural fields have been applied to a wide range of problems: • 3D scene reconstruction: neural fields can be used to model the properties of 3D scenes (i.e., geometry, appearance, materials, and lighting), in both static and dynamic cases. which provide an efficient and continuous representation of the geometry. Another example is represented by neural radiance fields (NeRFs), that learn to render 3D scenes, by mapping coordinates and viewing angles to the corresponding radiance and density. • Digital humans: neural fields can be used to model human shape and appearance and can include information on the complex movements of a human body. In this context, the ability of neural fields to model input and solution in a continuous and differentiable manner is invaluable. For example, physics-informed neural networks (PINNs) use neural fields to include, in the training objective, the residual computed via automatic differentiation. Instead, encode-process-decode architectures (e.g. CORAL), built on conditional neural fields, have been explored as an alternative operator-learning technique. ==See also==
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