物理约束 ML

Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler–Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE.

Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data-fidelity term, added at a few points in the domain to the standard residual and boundary losses. We show that this supervision term acts as an operator-level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, independently of how the pointwise labels are obtained. For a broad class of PDEs admitting a Feynman-Kac (FK) representation, we generate such labels by Monte Carlo averages of the FK functional, resulting in what we call "FK-PINNs", and using the excess risk decomposition approach, we derive non-asymptotic $L^2(\Omega)$-error bounds for FK-PINNs with $\tanh$ activation trained by finitely many steps of gradient descent. Along the way, we establish pseudo-dimension bounds for first- and second-order derivatives of $\tanh$ neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schrödinger, mean exit time, and committor problems corroborate the theory, and show that FK-PINNs can successfully solve PDEs for which standard PINNs exhibit severe failure modes.

Physics-Informed Neural Networks (PINNs) embed physical laws into deep learning models. However, conventional PINNs often suffer from failure modes leading to inaccurate solutions. We trace these failure modes to two structural pathologies: gradient shattering, where gradients degrade with depth and provide little training signal, and flow mismatch, where training pushes predictions along trajectories that diverge from the PDE solution path. We introduce ResPINNs, which reformulate PINNs as residual flows, networks that iteratively refine their own predictions through explicit corrective steps, in the spirit of classical iterative solvers. Our analysis shows that this design mitigates both pathologies by keeping updates aligned with descent and by preserving informative gradients across depth. Extensive experiments on PDE benchmarks confirm that ResPINNs achieve higher accuracy with substantially fewer parameters than conventional architectures.

Many fundamental biological processes are governed by mechanical forces, with proteins acting as the key molecular mediators. Elucidating how protein unfolding responds to force is critical for understanding the mechano-pathologies, such as cardiomyopathy and muscular dystrophy. While the unfolding trajectories measured by Single-Molecule Force Spectroscopy (SMFS) map the instantaneous force response against molecular extension, its broader application is limited by time-consuming data collection and high operational costs. Here, we present the first scalable generative diffusion framework for full unfolding trajectory prediction, which integrates protein encoders for multi-scale conditioning. Beyond establishing the field's first systematic benchmark using existing models, we propose GenUnfold, a novel physics-guided diffusion model that combines global coevolutionary context with a local mechanical representation of the protein. The representation is derived from a novel physics-biased attention mechanism, which steers the generative diffusion process by modeling dynamic residue dependencies as a function of both structural topology and interaction stiffness. The benchmark for this task is built upon the biomolecule stretching database and several representative baseline models. Empirical results demonstrate that GenUnfold achieves state-of-the-art performance, reducing distributional error (FID) by 30\% and 54\% compared to pretrained Evolutionary Scale Model (ESM)-2 and standard transformer, respectively. Beyond statistical curve similarity, GenUnfold demonstrates superior physical consistency; in downstream mechanical property prediction, it reduces prediction errors for unfolding force and energy distributions by 6\% and 36\% over the ESM-2 baseline. These results indicate that while existing generative AI approaches can alleviate the need for predicting representative force curves, GenUnfold further improves performance by leveraging the synergy between protein structure and evolutionary information. By enabling proteome-wide screening to identify mechanical candidates before costly physical validation, our approach is promising to accelerate the discovery of force-targeted therapeutics.

Current data-driven scientific modeling struggles with a functional dichotomy: neural operators exhibit spectral bias in high-frequency regimes, while physics-constrained paradigms suffer from optimization pathologies. To bridge this gap, we propose Adaptive Physics-Informed Computing (APIC), a neuro-symbolic meta-architecture designed with structural reconfigurability to encode diverse domain priors. Crucially, APIC integrates a gradient isolation strategy that reduces interference between the optimization paths of parameter identification and residual correction, effectively mitigating gradient conflicts. By instantiating this framework for nonlinear dissipative systems, we derive the Generalized Kuramoto-Sivashinsky-Cahn-Hilliard (G-KSCH) kernel, providing a unified representation for sparse dynamic identification. Extensive experiments demonstrate that APIC establishes new benchmarks in 3D compressible supersonic shock wave prediction, surpassing diverse architectures (e.g., CNNs and Transformers) by substantial margins in predictive accuracy. Notably, APIC achieves Pareto-optimal performance, delivering superior precision with reduced computational overhead compared to SOTA models, while exhibiting strong cross-task adaptability across meteorological and urban traffic datasets.

Physics-informed neural networks (PINNs) enforce physical laws by minimizing partial differential equation (PDE) residuals and auxiliary constraints. Standard training relies on a mean-squared error (MSE) objective, which implicitly assumes independent Gaussian residuals with a fixed global variance. We show theoretically and empirically that residuals encountered during PINN training are heterogeneous and heavy-tailed, revealing a systematic mismatch with this assumption. As a consequence, a small number of large residuals can disproportionately dominate both the loss and gradient, leading to poorly balanced optimization dynamics. Motivated by this mismatch, we adopt a Student-$t$ residual model to explicitly capture heavy-tailed behavior. An equivalent hierarchical representation yields an expectation–maximization (EM) algorithm that alternates between estimating residual-dependent weights and optimizing network parameters via a weighted MSE objective, allowing existing PINN solvers to be reused in the M-step. The resulting training dynamics bound the influence of extreme residuals and admit almost sure convergence guarantees under standard stochastic optimization assumptions. Experiments across a diverse suite of challenging PDE benchmarks demonstrate consistently improved solution accuracy and robustness compared to standard PINN training.

Diffusion models show growing promise for generative modeling of physical systems, but enforcing partial differential equation (PDE) constraints directly is infeasible during the stochastic denoising process. Current methods apply constraints to the expected clean sample, incurring a Jensen’s Gap that forces a trade-off between PDE satisfaction and generative accuracy. To bridge this gap, we propose Physics-Informed Distillation of Diffusion Models (PIDDM), a simple yet effective post-hoc distillation strategy that enforces PDE constraints after training. PIDDM enables fast single-step generation while improving both physical consistency and sample quality, supporting forward/inverse problems and reconstruction from partial observations. Extensive experiments across PDE benchmarks show PIDDM outperforms recent baselines, such as PIDM, DiffusionPDE, and ECI-sampling, in both accuracy and constraint satisfaction, with lower computation and minimal hyperparameter tuning, offering a more efficient pathway to physics-informed diffusion models.

Machine-learned (ML) XC functionals aim to replace human-designed density functional approximations by learning directly from reference data, but they still do not consistently outperform traditional $\mathcal{O}(N^4)$-scaling hybrid functionals. We therefore study a hybrid-distillation setting, where $\mathcal{O}(N^3)$-scaling semilocal ML-XC functionals are trained to reproduce B3LYP/def2-SVP targets. We introduce Derivative Informed XC-Loss (DI-Loss), a loss that incorporates additional information from the reference hybrid functional by supervising first and second derivatives of the energy on the Grassmannian of admissible density matrices. Rather than only matching the self-consistent fixed point, DI-Loss aligns the local first- and second-order response of the learned functional with that of the target functional. Across four evaluated architectures, DI-Loss consistently improves the main energy metrics. Averaged uniformly across architectures, the total-energy MAE decreases by 66% relative to energy and density supervision alone. The density-sensitive mean-field energy metric $E_\rho$ improves from 1.2 to 0.8 mEh on average, while dipole and $\mathcal{L}_2$ density errors do not improve uniformly. We further show that densities from the distilled functionals reduce hybrid-functional SCF iterations by up to 55%. In downstream TDDFT calculations, Hessian supervision improves excited-state predictions, with XCdiff reducing the mean excitation-energy MAE by 24-35% across molecule sizes on QM40.

Riemannian diffusion models generalize score-based generative modeling to manifold-supported data via stochastic diffusion equations on the manifold. However, training requires sampling from and differentiating the manifold heat kernel, which is rarely available in closed form beyond a few highly symmetric manifolds. We propose a general approach that approximates the heat kernel by directly solving the manifold heat equation with a physics-informed neural network (PINN). Given an explicit manifold specification, we choose a coordinate system, derive the corresponding heat (Fokker--Planck) equation and a short-time asymptotic approximation, and then train a PINN to learn the log heat kernel. The resulting surrogate enables both forward noising (heat-kernel sampling) and conditional-score evaluation for denoising score matching. We demonstrate the method on diverse manifolds including $S^2$, $SO(3)$, $\mathrm{SPD}(n)$, and permutation-quotiented point clouds.

Depth estimation for transparent objects remains a fundamental challenge, as RGB-based cues often fail in regions affected by refraction and light transmission. Polarization provides physically grounded information related to surface orientation and material properties, offering reliable geometric cues even in the absence of texture. In this work, we introduce PolarDepth, a monocular framework that incorporates both RGB and polarization inputs, including the degree and angle of linear polarization (DoLP and AoLP), to estimate dense depth and localize transparent regions. PolarDepth injects polarization-derived physical priors by estimating the refractive index, zenith angle, and azimuth angle from polarization measurements and embedding them into an implicit geometric representation that constrains depth inference in ambiguous transparent regions. To support model development and evaluation, we introduce PTOD, a dataset with synchronized RGB, polarization, and depth data and manually annotated transparent region masks. Experimental results demonstrate that PolarDepth achieves state-of-the-art performance in transparent object depth estimation. The findings highlight the effectiveness of embedding polarization-derived physical priors into learned representations for robust perception in complex visual environments.

Partial Differential Equations (PDEs) play a fundamental role in scientific computing, and recent efforts have sought to extend the success of foundation models to PDE solving. However, multi-physics PDE pre-training faces the unique challenge of disentangling dynamic heterogeneity to learn universal, elementary patterns that generalize to new PDEs. Additionally, cross-physics transfer lacks a theoretical framework for interpretability—specifically, understanding which pre-trained operator knowledge is effectively transferred to target PDEs. To bridge these gaps, we introduce the theory of neural operator splitting, which decomposes PDE evolution into a modulated global spectral operator and sparse local constitutive mechanisms. A key innovation is Origo, which provides a neural operator bank that enables the identification of operator-level generalization patterns. Extensive experiments demonstrate strong zero-shot generalization and mechanism-level interpretability on unseen PDEs.

Physics-informed neural networks (PINNs) have emerged as a powerful deep learning approach for solving partial differential equations (PDEs) in the physical sciences, yet their behavior remains largely opaque and is typically understood through failure mode analyses rather than explicit interpretability. To address this issue, we introduce PINNfluence, a training data attribution framework for interpreting PINNs based on influence functions. By extending influence functions to composite physics-informed training objectives, we enable fine-grained attribution between predictions, loss components, and training data points. Through benchmark experiments across various PDEs, we demonstrate that influence patterns provide granular diagnostics that distinguish structural characteristics across well-trained and poorly-trained PINNs. PINNfluence thus opens a new avenue for understanding and improving the reliability of PINNs through the lens of their data.

In open-world intelligent systems, processing continuous sensory streams disrupted by heterogeneous degradation sources presents a fundamental challenge: reconciling the inherent tension between observational completeness and reconstruction fidelity. Methods that prioritize completeness by bridging long-term occlusions often introduce spurious artifacts, while approaches that focus on aggressive noise suppression inevitably disrupt temporal continuity and erase valid structures. To address this challenge, we propose NeuroMamba, a universal plug-and-play module that enhances spatiotemporal consistency in degraded streams. NeuroMamba tackles the dual objectives through two synergistic components. First, we introduce a regional Hybrid Spatiotemporal Rectification (HSR) module, which leverages Mamba-based inertial modeling of linear complexity to recover short-horizon temporal dependencies and infer missing modalities under partial observability. Second, we design a Spiking Confidence Gate (SCG) that enforces reconstruction fidelity under occupancy-guided supervision. Implemented as a hard-thresholding spiking gate unit based on leaky integrate-and-fire (LIF) neurons, SCG distinguishes valid geometric features from sensor noise based on accumulated temporal evidence. Extensive experiments on the nuScenes robustness benchmark demonstrate that NeuroMamba effectively reconciles the trade-off between completeness and fidelity, outperforming the performance of existing approaches in restoring high-fidelity spatiotemporal features from severely incomplete and degraded observations.

Mental rotation—the ability to compare objects seen from different viewpoints—is a fundamental example of mental simulation and spatial world modeling in humans. Here we propose a mechanistic model of human mental rotation, leveraging recent advances in deep, equivariant, and neuro-symbolic learning. Our model consists of three stacked components: (1) an equivariant neural encoder, producing 3D spatial representations of objects from images, (2) a neuro-symbolic object encoder, deriving symbolic objects descriptions from these spatial representations, and (3) a neural decision agent, comparing these symbolic descriptions to prescribe rotation simulations in 3D latent space via a recurrent pathway. Our model design is guided by the existing experimental literature on mental rotation, which we complemented with experiments in VR where participants could at times manipulate the objects to compare. Our model captures well the performance, response times and behavior of participants in our and others' experiments, and through ablation studies we demonstrate the necessity of each component. Our work adds to a recent collection of deep neural models of human spatial reasoning, further demonstrating the potency of integrating deep, equivariant, and symbolic representations to model the human mind.

In multi-view clustering (MVC), conventional anchor learning based models implicitly assume a uniform distribution of anchors across clusters, which could lead to inferior representation, especially when clusters vary significantly in size, as larger clusters require more anchors so as to adequately capture their intrinsic structural complexity. To alleviate this, we design a method termed FCFMVC that explicitly encourages proportional anchor allocation. To be specific, we transfer anchor allocation to discrete sample-cluster learning via bipartite graph bridge, and then backpropagate cluster state consisting of size and dispersion degree to guide anchor assignment. This allows the model to integrate cluster cardinality awareness and structural compactness directly into anchor distribution. On the other hand, we regard anchors as pseudo-samples, introduce an anchor-cluster indicator matrix on each view, and directly constrain the number of anchors assigned to each cluster within a tolerance margin. These two paths are further coupled through anchor-sample label alignment, and collaboratively facilitate anchor generation from fine-grained (anchor-level) to coarse-grained (cluster-level) structures. Besides, the entire optimization operation with linear time and space cost makes FCFMVC well-scalable to large-scale tasks. Experiments on datasets with diverse scales confirm the effectiveness of our FCFMVC.

While neural PDE solvers have demonstrated significant potential for accelerating engineering simulations, existing architectures remain constrained by high memory consumption and the "single-node bottleneck," where the maximum processable mesh resolution is strictly limited by the VRAM of a single compute unit. To address these challenges, we propose PGD-NO, a neural operator with Precomputed Geometry Decomposition, that relocates the computational overhead of geometric encoding to a deterministic pre-computation phase. By utilizing an iterative geometry decomposition algorithm to extract "geometry tokens," our model decouples feature extraction from solution querying. This architecture enables linear memory scalability, allowing high-fidelity learning on meshes exceeding 10 million nodes—a scale where existing architectures typically encounter memory exhaustion. PGD-NO demonstrates competitive predictive accuracy across diverse industrial benchmarks and provides intrinsic interpretability through attention mechanisms. By effectively overcoming traditional mesh-size constraints, PGD-NO offers a robust and efficient solution for the next generation of large-scale, high-fidelity industrial design applications.

Physics-informed neural networks (PINNs) can be unreliable on PDEs with oscillatory, multiscale, stiff, or long-time solutions, and these difficulties worsen in high dimensions where collocation-based training yields large numerical integration error and high-variance gradients. We propose Low-Rank Fourier Sums (LoRFS), representing the solution as a low-rank sum of separable Fourier expansions (products of one-dimensional Fourier series across coordinates). This makes high-frequency structure explicit and enables closed-form evaluation of common physics-based objectives and their gradients (e.g., $L^2$ residual and variational losses), replacing sampling-based collocation estimates with analytic loss evaluation and eliminating sampling noise. We further provide theoretical results that clarify why LoRFS is particularly well suited to high-dimensional regimes. Across canonical PINN failure-mode benchmarks and their high-dimensional extensions, LoRFS consistently outperforms strong PINN baselines and remains stable in regimes where competing methods degrade.

We study the error rate of LLMs on tasks like arithmetic that require a deterministic output, and repetitive processing of tokens drawn from a small set of alternatives. We argue that incorrect predictions arise when small errors in the attention mechanism accumulate to cross a threshold, and use this insight to derive a quantitative two-parameter relationship between the accuracy and the complexity of the task. The two parameters vary with the prompt and the model; they can be interpreted in terms of an elementary noise rate, and the number of plausible erroneous tokens that can be predicted. Our analysis is inspired by an "effective field theory'' perspective: the LLM's many raw parameters can be reorganized into just two parameters that govern the error rate. We perform extensive empirical tests, using Gemini 2.5 Flash, Gemini 2.5 Pro and DeepSeek R1, and find excellent agreement between the predicted and observed accuracy for a variety of tasks, although we also identify deviations in some cases. Our model provides an alternative to suggestions that errors made by LLMs on long repetitive tasks indicate the "collapse of reasoning'', or an inability to express "compositional'' functions. Finally, we show how to construct prompts to reduce the error rate.

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce *LieStoNet*, an end-to-end, template-free framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics. Code is available at this link.

Modern deep learning for asset allocation typically separates forecasting from optimization. We argue this creates a fundamental mismatch where minimizing prediction errors fails to yield robust portfolios. We propose the Signature Informed Transformer to address this by unifying feature extraction and decision making into a single policy. Our model employs path signatures to encode complex path dependencies and introduces a specialized attention mechanism that targets geometric asset relationships. By directly minimizing the Conditional Value at Risk we ensure the training objective aligns with financial goals. We prove that our attention module rigorously amplifies signature derived signals. Experiments across diverse equity universes show our approach significantly outperforms both traditional strategies and advanced forecasting baselines. The code is available at: https://github.com/Yoontae6719/Signature-Informed-Transformer-For-Asset-Allocation