Abstract
Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high-frequency functions. This observation identifies two important factors in resolving rapid oscillations: the initial slope scale and the distribution of partition points induced by the networks. Motivated by this analysis, we propose RepNN, a reparameterized neural network model with activation ReLU or tanh designed for high-frequency and multiscale problems. The key idea is to reparameterize the weights and biases in the first hidden layer, which enables effective control of the initial slope scale and provides an appropriate distribution of the initial partition points. Furthermore, treating the reparameterized weights and biases as trainable parameters allows the DNN to achieve adaptive frequency scaling during training. In addition, we derive quantitative estimates for the output and slope magnitudes of the reparameterized DNN to guide the initialization of the proposed method. Numerical experiments, including multiscale one- and four-dimensional function approximations, forward and inverse PDE problems in combination with physics-informed neural networks (PINNs), and operator learning for an earthquake problem using real data, demonstrate that RepNN improves the predicted accuracy of vanilla DNNs in capturing highly oscillatory features with slightly additional computational cost. These results indicate that RepNN provides an effective and flexible approach for overcoming spectral bias and applying DNNs to multiscale problems.
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The architecture of a neural network and the choice of its activation function are both fundamental to its performance. Equally important is ensuring that these two elements are well matched, as their alignment is key to effective representation and learning. In this paper, we introduce the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN), a model that combines sine-type activations with the multi-component and multi-layer structure of MMNNs. In an FMMNN, each component is represented as a trainable linear combination of fixed random sine-type basis functions, while multi-layer composition generates more complex and adaptive high-frequency features. We establish that FMMNNs retain exponential expressive power for function approximation even under a low-rank architectural structure. We also analyze the optimization landscape of FMMNNs and find it to be substantially more favorable than that of standard fully connected neural networks, especially for high-frequency targets. In addition, we propose a scaled random initialization method for the first-layer weights in FMMNNs, which accelerates training and improves final performance when sufficient samples are available. Extensive numerical experiments support our theoretical insights, showing that FMMNNs achieve strong accuracy and favorable convergence behavior on oscillatory function-approximation benchmarks.
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Implicit Neural Representations (INRs) have been proven successful in encoding continuous signals through coordinate-based networks, yet facing a spectral dilemma: periodic activations capture fine details but act as all-pass filters that memorise noise, while spatially compact activations regularise effectively but suffer from low-frequency bias. Existing attempts to resolve this trade-off introduce computational overhead or tuning frailty. We propose to model each neuron's activation as the steady-state response of a sinusoidally-forced damped harmonic oscillator, whose amplitude naturally governs the network's spectral selectivity during training. By jointly optimising the oscillator parameters alongside the network weights, our method adapts to the target signal's spectral content without explicit regularisation. Initialised in the stopband, the network exhibits a coarse-to-fine learning curriculum that progressively expands its spectral gate, capturing low-frequency structures first and high-frequency details only when justified by the reconstruction objective. Comprehensive experiments show that our approach consistently achieves state-of-the-art or competitive results against established INRs, while requiring no task-specific tuning of any hyperparameters.
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