complex-valued neural networks

Existing neural operators struggle to efficiently model parametric and coupled partial differential equations (PDEs). This work addresses this limitation by extending the Fourier Neural Operator (FNO) with minimal architectural modifications: it introduces a hypernetwork-driven, parameter-aware modulation mechanism to condition the operator on physical parameters, and systematically designs an operator structure for coupled PDEs that balances shared representations with cross-variable interactions. The resulting approach significantly improves modeling accuracy while preserving computational efficiency. On benchmark problems including capacitively coupled plasma and the Gray–Scott system, the method reduces prediction errors by 55%–72% compared to strong baselines.

Modeling periodic and aperiodic functions in scientific machine learning remains challenging due to limitations of existing Fourier-based neural networks—particularly their reliance on harmonic constraints and inability to decouple dimensional dependencies. Method: This paper proposes the Fourier Learning Machine (FLM), a novel neural architecture grounded in non-harmonic Fourier series, employing cosine activation units with learnable frequencies, amplitudes, and phases. Under multidimensional separability, FLM enables feedforward spectral basis representation and establishes, for the first time, a bijective mapping between Fourier coefficients and phase-amplitude parameters, supporting equivalent conversion between full spectral bases and phase-shifted forms. Contribution/Results: FLM overcomes harmonic and dimensional coupling restrictions inherent in conventional Fourier networks. Experiments demonstrate that FLM matches or surpasses SIREN and standard MLPs in solving partial differential equations and optimal control problems, validating its efficiency, generalization capability, and expressive power for scientific computing tasks.

Standard multilayer perceptrons (MLPs) suffer from spectral bias, hindering accurate modeling of high-frequency complex-valued optical field perturbations. To address this, we propose Fourier Feature Networks (FFNs), which map inputs into a perturbation-dependent Fourier basis space, transforming nonlinear learning into linear combination of precomputed basis functions. FFNs enable end-to-end learning of the complex-valued transmission matrix under multimode fiber compression. This approach significantly reduces model complexity while enhancing generalization. Experiments demonstrate that FFN achieves one-order-of-magnitude lower prediction error than standard MLPs, attains an average complex correlation coefficient of 0.995 for both amplitude and phase, and reduces parameter count by 85%. By explicitly encoding high-frequency priors via Fourier features, FFN effectively overcomes the representational bottleneck of conventional neural networks in optical high-frequency modeling.

This study investigates under what conditions complex-valued neural networks (CVNNs) genuinely outperform their real-valued counterparts, clarifying the relationship between input complex structure and learning advantages. Employing a representation-first evaluation framework, the authors systematically compare CVNNs against parameter- and FLOP-matched real-valued baselines across radio frequency, quantum wavefunction, and EEG tasks, complemented by analyses involving polar/Cartesian coordinate transformations, CReLU activation, gradient dynamics, and hyperparameter ablation. The findings reveal that CVNNs’ superiority stems from specific representational structures, symmetries, and optimization properties—not inherent architectural dominance—and is most pronounced in PSK modulation tasks, whereas magnitude-based real models excel in QAM. After independent hyperparameter tuning, CVNNs achieve only a 2.46 percentage point gain on RadioML, substantially lower than the previously reported 22.94, indicating that performance gaps are largely attributable to hyperparameter sensitivity.

Modeling, classification, and forecasting of large-scale spatiotemporal data from high-dimensional nonlinear complex systems—such as brain activity, climate, and ecosystems—remain challenging due to the limited representational capacity of conventional dimensionality reduction and phase-space reconstruction methods. Method: We propose a geometric vector field analysis framework on discrete measure spaces, introducing for the first time a two-parameter family of vector field metrics applicable to spatiotemporal functions defined on graphs and simplicial complexes. This framework unifies representations of scalar fields, gradient fields, and multivalued fields, transcending classical attractor-geometric limitations. By integrating vector field representation theory, discrete differential geometry, and multidimensional scaling (MDS), it enables model-free, efficient dimensionality reduction, modal decomposition, phase-space reconstruction, and attractor characterization. Results: Extensive validation on biological and physical simulation datasets demonstrates substantial improvements in dynamical system analysis capability, particularly in capturing nonlinear, multiscale spatiotemporal structures.

This work addresses the limitation of existing Fourier Neural Operators (FNOs) in effectively modeling structured couplings among Fourier modes in nonlinear partial differential equations. To overcome this, the authors propose a Higher-Order Spectral Convolution mechanism (HO-FNO), which explicitly introduces multilinear mode mixing in the frequency domain. This approach generalizes FNO’s diagonal modulation to higher-order nonlinear interactions, embedding inductive biases aligned with the dynamics of nonlinear PDEs. Empirical results demonstrate that HO-FNO significantly outperforms current spectral neural operators across multiple benchmarks: a single-layer HO-FNO surpasses a 16-layer FNO in highly nonlinear regimes and matches or exceeds the performance of advanced Transformers and state-space models.

This study systematically investigates the frequency-domain encoding capabilities of the Chronos foundation model, addressing a critical gap in understanding how such models represent fundamental signal properties. Through controlled experiments using discrete sinusoidal signals and a lightweight online Minimum Description Length (MDL) probing framework, the work examines the existence, separability, and cross-spectral fidelity of internal frequency representations within the Chronos decoder. The research reveals, for the first time, a degradation in representation quality in high-frequency regions, thereby delineating both the strengths and limitations of Chronos’s frequency encoding mechanism. These findings offer novel insights into the interpretability of time-series foundation models and provide practical guidance for applications in signal processing and multimodal fusion.

This work addresses the limited generalization capability of existing methods in complex real-world scenarios by proposing a novel framework based on adaptive feature fusion and dynamic inference. The approach enhances model robustness under distribution shifts through multi-level semantic alignment and an uncertainty-aware module. Extensive experiments demonstrate that the proposed method significantly outperforms state-of-the-art models across multiple benchmark datasets, achieving an average accuracy improvement of 3.2% while maintaining low computational overhead. This study offers a promising technical pathway toward building reliable artificial intelligence systems capable of operating effectively in open and dynamically changing environments.

This work addresses the challenge of efficiently modeling periodicity in multivariate time series forecasting, where existing methods often fail to explicitly capture periodic patterns and neglect the intrinsic coupling between phase and amplitude. To overcome these limitations, the authors propose a lightweight, period-aware phase-amplitude modulation network that explicitly decouples and interactively models phase shifts and amplitude variations through a dual-branch architecture. The approach introduces learnable cyclic phase embeddings and element-wise amplitude modulation mechanisms, thereby avoiding computationally expensive attention structures. Extensive experiments on twelve real-world datasets demonstrate state-of-the-art performance, validating the effectiveness and superiority of explicitly disentangling phase and amplitude for periodic modeling in time series forecasting.