Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks

The numerical solution of differential equations using neural networks has become a central topic in scientific computing, with Physics-Informed Neural Networks (PINNs) emerging as a powerful paradigm for both forward and inverse problems. However, unlike classical numerical methods that offer established convergence guarantees, neural network-based approximations typically lack rigorous error bounds. Furthermore, the non-deterministic nature of their optimization makes it difficult to mathematically certify their accuracy. To address these challenges, we propose a "Learn and Verify" framework that provides computable, mathematically rigorous error bounds for the solutions of differential equations. By combining a novel Doubly Smoothed Maximum (DSM) loss for training with interval arithmetic for verification, we compute rigorous a posteriori error bounds as machine-verifiable proofs. Numerical experiments on nonlinear Ordinary Differential Equations (ODEs), including problems with time-varying coefficients and finite-time blow-up, demonstrate that the proposed framework successfully constructs rigorous enclosures of the true solutions, establishing a foundation for trustworthy scientific machine learning.

Construction and Evaluation of a Self-Attention Model for Semantic Understanding of Sentence-Final Particles

Sentence-final particles serve an essential role in spoken Japanese because they express the speaker's mental attitudes toward a proposition and/or an interlocutor. They are acquired at early ages and occur very frequently in everyday conversation. However, there has been little proposal for a computational model of acquiring sentence-final particles. This paper proposes Subjective BERT, a self-attention model that takes various subjective senses in addition to language and images as input and learns the relationship between words and subjective senses. An evaluation experiment revealed that the model understands the usage of "yo", which expresses the speaker's intention to communicate new information, and that of "ne", which denotes the speaker's desire to confirm that some information is shared.