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Lucas Nissenbaum

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AAAI Conference 2025 Conference Paper

Neural Conjugate Flows: A Physics-Informed Architecture with Flow Structure

  • Arthur Bizzi
  • Lucas Nissenbaum
  • João M. Pereira

We introduce Neural Conjugate Flows (NCF), a class of neural-network architectures equipped with exact flow structure. By leveraging topological conjugation, we prove that these networks are not only naturally isomorphic to a continuous group, but are also universal approximators for flows of ordinary differential equation (ODEs). Furthermore, topological properties of these flows can be enforced by the architecture in an interpretable manner. We demonstrate in numerical experiments how this topological group structure leads to concrete computational gains over other physics informed neural networks in estimating and extrapolating latent dynamics of ODEs, while training up to five times faster than other flow-based architectures.

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NeurIPS Conference 2025 Conference Paper

Neuro-Spectral Architectures for Causal Physics-Informed Networks

  • Arthur Bizzi
  • Leonardo Moreira
  • Márcio Marques
  • Leonardo Mendonça
  • Christian de Oliveira
  • Vitor Balestro
  • Lucas dos Santos Fernandez
  • Daniel Yukimura

Physics-Informed Neural Networks (PINNs) have emerged as a powerful frame- work for solving partial differential equations (PDEs). However, standard MLP- based PINNs often fail to converge when dealing with complex initial value problems, leading to solutions that violate causality and suffer from a spectral bias towards low-frequency components. To address these issues, we introduce NeuSA (Neuro-Spectral Architectures), a novel class of PINNs inspired by classi- cal spectral methods, designed to solve linear and nonlinear PDEs with variable coefficients. NeuSA learns a projection of the underlying PDE onto a spectral basis, leading to a finite-dimensional representation of the dynamics which is then integrated with an adapted Neural ODE (NODE). This allows us to overcome spectral bias, by leveraging the high-frequency components enabled by the spectral representation; to enforce causality, by inheriting the causal structure of NODEs, and to start training near the target solution, by means of an initialization scheme based on classical methods. We validate NeuSA on canonical benchmarks for lin- ear and nonlinear wave equations, demonstrating strong performance as compared to other architectures, with faster convergence, improved temporal consistency and superior predictive accuracy. Code and pretrained models are available in https: //github. com/arthur-bizzi/neusa.

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