Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2507.23141

AI paradigm for solving differential equations: first-principles data generation and scale-dilation operator AI solver

Xiangshu Gong·Zhiqiang Xie·Xiaowei Jin·Chen Wang·Yanling Qu·Wangmeng Zuo·Hui Li
View original source ↗Read source document ↗Export metadata ↓

Abstract

Many problems are governed by differential equations (DEs). Artificial intelligence (AI) is a new path for solving DEs. However, data is very scarce and existing AI solvers struggle with approximation of high frequency components (AHFC). We propose an AI paradigm for solving diverse DEs, including DE-ruled first-principles data generation methodology and scale-dilation operator (SDO) AI solver. Using either prior knowledge or random fields, we generate solutions and then substitute them into the DEs to derive the sources and initial/boundary conditions through balancing DEs, thus producing arbitrarily vast amount of, first-principles-consistent training datasets at extremely low computational cost. We introduce a reversible SDO that leverages the Fourier transform of the multiscale solutions to fix AHFC, and design a spatiotemporally coupled, attention-based Transformer AI solver of DEs with SDO. An upper bound on the Hessian condition number of the loss function is proven to be proportional to the squared 2-norm of the solution gradient, revealing that SDO yields a smoother loss landscape, consequently fixing AHFC with efficient training. Extensive tests on diverse DEs demonstrate that our AI paradigm achieves consistently superior accuracy over state-of-the-art methods. This work makes AI solver of DEs to be truly usable in broad nature and engineering fields.

Explore related subjects

cs.LGphysics.comp-ph

Keep this discovery

BibTeXRIS

Xiangshu Gong, Zhiqiang Xie, Xiaowei Jin, Chen Wang, Yanling Qu, Wangmeng Zuo, Hui Li. 2025-07-30. AI paradigm for solving differential equations: first-principles data generation and scale-dilation operator AI solver. https://arxiv.org/abs/2507.23141

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

AUC Maximization from Biased Positive-unlabeled Data with Confidence

Maximizing the area under the receiver operating characteristic curve (AUC) is a standard approach to imbalanced binary classification. Although positive and negative data are required for maximizing the AUC, negative data are often difficult to collect in some real-world applications due to privacy concerns or the need for specialized expertise to annotate them. Thus, AUC maximization from positive and unlabeled (PU) data has been attracting attention. Existing methods assume that labeled positive data are unbiased samples from the true positive distribution. However, this ideal assumption is often violated in practice. In this paper, we propose a method to maximize the AUC from biased PU data. To address the bias, our key idea is to exploit {\it confidence}, i.e., the probability that an instance is positive, associated with the small number of labeled positive data. We derive an estimator of the AUC risk using biased PU data with confidence, enabling AUC maximization under such bias. We further show that the rewritten AUC risk induces a Bayes-optimal AUC ranking even when the available confidence is any strictly increasing transformation of the true posterior probability. We experimentally show the effectiveness of our method on eight real-world datasets.

cs.LG↗
arXiv2026

Measuring the Value of World-Model Updates: A Counterfactual Utility Protocol for Continual Adaptation

Continual world models must decide whether new data justify changing the model. Fixed replay schedules and prediction-error triggers specify when to update, but neither reveals the value of an individual update: one deployment run cannot show how the same model would have performed at that moment had it held its parameters. We introduce the fork ledger, which branches a deployment stream at pre-registered decision points into matched update and hold continuations under common random numbers. It evaluates both continuations on the same episodes and records $\Delta R = R_{\mathrm{update}} - R_{\mathrm{hold}}$. Always applying one fixed update mechanism lowers return on all three simulated control tasks: CartPole ($-144.0$; checkpoint-bootstrap $95\%$ CI $[-185.4,-116.1]$, against a converged return near $650$), Walker ($-82.8$; $[-101.1,-61.7]$) and Cheetah ($-18.6$; $[-29.0,-6.6]$). Divergence is an outcome of applying the update, so the estimand counts every attempted fork; restricted to the $693$ of $720$ that did not collapse, CartPole and Walker are unchanged in sign ($-113.4$ and $-82.1$) and Cheetah becomes unresolved ($-3.9$; $[-17.5,+13.0]$). The task is the unit of inference: each contributes $240$ attempted forks over five pretrained checkpoints crossed with two drift directions. The ledger makes counterfactual utility observable for a fixed mechanism, allowing triggers to be judged by the updates they select rather than by surprise detection alone.

cs.LG↗
arXiv2026

When More Is Not Better: Component Anti-Synergy in a P300 Speller

P300 brain-computer interface (BCI) spellers can provide hands-free communication for people with severe motor impairments. Modern pipelines combine multiple individually promising components, often assuming that 'more-is-better'. We tested this assumption using a four-component full-factorial experiment varying the inclusion of Euclidean Alignment (EA), xDAWN spatial filtering, subject calibration, and language model priors on a public P300 dataset. Performance was evaluated using accuracy, repetitions, and information transfer rate (ITR) with mixed-effects models. Results show that the value of components is conditional rather than additive. Calibration was the strongest singular contributor, while EA compensated for its absence in zero-calibration settings. Adding independently useful components could also reduce performance, revealing component anti-synergy. Contrary to conventional wisdom, LM support was not universally beneficial: its effect depends strongly on the strength of the underlying EEG pipeline, while results from a larger LM showed a similar pattern. Together, these findings challenge maximal 'all-on' pipeline design and highlight the value of selecting spatial and language-support components according to the quality of available EEG evidence.

cs.LG↗