Shun Sato

Daisuke Furihata, Shun Sato, Takayasu Matsuo

We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of difference operators is essential to the discrete conservation law. Unfortunately, however, when we employ the standard central difference operator, the simplest one, the numerical solutions often suffer from undesirable spatial oscillations. In this letter, we propose a novel "average-difference method," which is tougher against such oscillations, and combine it with an existing conservative method. Theoretical and numerical analysis in the linear case show the superiority of the proposed method.

Shun Sato, Takayasu Matsuo

Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper, we aim at the unified framework of numerical methods for such PDEs. However, due to the presence of the mixed derivative, we cannot discuss numerical methods without some appropriate reformulation, which is mathematically challenging itself. Therefore, we first propose a novel procedure for the reformulation of target PDEs into a standard form of evolutionary equations. This contribution may become an important basis not only of numerical analysis, but also of PDE-theory. In order to illustrate this point, we establish the global well-posedness of the sine-Gordon equation. After that, we classify and discuss the spatial discretizations based on the proposed reformulation technique. As a result, we show the average-difference method is suitable for the discretization of the mixed derivative.

Kaito Sato, Shun Sato, Takayasu Matsuo

Multi-symplectic diamond schemes proposed by McLachlan and Wilkins (2015) provide a framework for the numerical integration of Hamiltonian partial differential equations, combining local implicitness with high-order accuracy and discrete multi-symplectic conservation laws. Despite these advantages, their behavior beyond a limited class of model equations remains poorly understood, and numerical difficulties may arise depending on the underlying multi-symplectic formulation. In this paper, we present a systematic stability analysis framework for diamond schemes applied to general multi-symplectic PDEs. The approach consists of three stages. First, we identify structural inconsistency of the local diamond update using Dulmage--Mendelsohn decomposition, revealing cases in which the scheme is intrinsically unsolvable. Second, we introduce a graph-based error-propagation analysis that yields a necessary stability condition by detecting negative cycles in a weighted directed graph. Third, for equations that pass the preliminary tests, we derive eigenvalue-based timestep restrictions providing sufficient conditions for stability. The analysis leads to a comprehensive classification of multi-symplectic PDEs according to whether diamond schemes are structurally inconsistent, unconditionally unstable, or conditionally stable. In particular, we show that benchmark equations such as the Korteweg--de Vries equation are intrinsically incompatible with the diamond update, while systems including the nonlinear Dirac and ``good'' Boussinesq equations admit stability regimes under mild timestep scaling. Extensive numerical experiments confirm the theoretical predictions and demonstrate the practical implications of the proposed framework. Our results clarify fundamental limitations of diamond schemes and provide practical guidelines for their reliable application to new PDE models.

Shun Sato

The modified Hunter--Saxton equation models the propagation of short capillary-gravity waves. As it involves a mixed derivative, its initial value problem on the periodic domain is much more complicated than the standard evolutionary equations. Although its local well-posedness has recently been proved, the behavior of its solution is yet to be investigated. In this paper, to develop a reliable numerical method for this problem, we derive a conservative finite difference scheme. Then, we rigorously prove not only its stability in the sense of the uniform norm but also its uniform convergence to sufficiently smooth exact solutions. Discrete conservation laws are used to overcome the difficulty due to the mixed derivative.

Shun Sato

In this paper, we consider the use of discrete gradients for differential-algebraic equations (DAEs) with a conservation/dissipation law. As one of the most popular numerical methods for conservative/dissipative ordinary differential equations, the framework of discrete gradient methods has been intensively developed over recent decades. Although discrete gradients have been applied to several specific conservative/dissipative DAEs, no unified framework for DAEs has yet been constructed. In this paper, we move toward the establishment of such a framework, and introduce concepts including an appropriate linear gradient structure for DAEs. Then, we reveal that the simple use of discrete gradients does not imply the discrete conservation/dissipation laws. Fortunately, however, we can successfully construct a new discrete gradient method for the case of index-1 DAEs. We believe this first attempt provides an indispensable basis for constructing a unified framework of discrete gradient methods for DAEs.

Shun Sato, Issei Sato

Symbolic regression aims to discover mathematical equations that fit given numerical data. It has been applied in various fields of scientific research, such as producing human-readable expressions that explain physical phenomena. Recently, Neural symbolic regression (NSR) methods that involve Transformers pre-trained on large-scale synthetic datasets have gained attention. While these methods offer advantages such as short inference time, they suffer from low performance, particularly when the number of input variables is large. In this study, we hypothesized that this limitation stems from the memorization bias of Transformers in symbolic regression. We conducted a quantitative evaluation of this bias in Transformers using a synthetic dataset and found that Transformers rarely generate expressions not present in the training data. Additional theoretical analysis reveals that this bias arises from the Transformer's inability to construct expressions compositionally while verifying their numerical validity. We finally examined if tailoring test-time strategies can lead to reduced memorization bias and better performance. We empirically demonstrate that providing additional information to the model at test time can significantly mitigate memorization bias. On the other hand, we also find that reducing memorization bias does not necessarily correlate with improved performance. These findings contribute to a deeper understanding of the limitations of NSR approaches and offer a foundation for designing more robust, generalizable symbolic regression methods. Code is available at https://github.com/Shun-0922/Mem-Bias-NSR .

Sewon Min, Jordan Boyd-Graber, Chris Alberti et al.

We review the EfficientQA competition from NeurIPS 2020. The competition focused on open-domain question answering (QA), where systems take natural language questions as input and return natural language answers. The aim of the competition was to build systems that can predict correct answers while also satisfying strict on-disk memory budgets. These memory budgets were designed to encourage contestants to explore the trade-off between storing retrieval corpora or the parameters of learned models. In this report, we describe the motivation and organization of the competition, review the best submissions, and analyze system predictions to inform a discussion of evaluation for open-domain QA.