A hybrid neural-spectral method constructs the first fully non-linear rotating black hole solutions in cubic Lovelock gravity, parametric in the gravitational coupling constants.
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Discovery of unstable singularities
15 Pith papers cite this work. Polarity classification is still indexing.
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Constructs C^α self-similar blowup profiles for 3D Euler vorticity without swirl and proves asymptotically self-similar blowup from C_c^α data, with limiting factorization as α→(1/3)^-.
ASYS recovers known analytical PDE forms and generates new interpretable symbolic approximations, such as a geometric interface formula for 2D Allen-Cahn and a nine-parameter contraction law for Keller-Segel blow-up, via agent-guided evolutionary search on differentiable programs.
PINNs approximate near-minimal surfaces bounding knots in S^3; their self-intersection numbers align with Fine's conjecture predictions derived from the HOMFLY polynomial.
For every d≥3 and every α<1−2/d, axisymmetric swirl-free incompressible Euler admits self-similar blow-up solutions with C^{1,α} initial velocity that is smooth away from the origin.
PINNs trained with incorrect PDE parameters achieve losses matching or beating the correct baseline while their solutions differ by up to 128%.
Numerical construction of unstable self-similar axially symmetric swirl-free solutions to the incompressible Navier-Stokes equations on R^3 with global pointwise residuals of order 10^{-10}.
beignet replaces random Fourier feature embeddings in PINNs with a trainable multi-resolution Fourier feature pyramid, achieving higher accuracy on PDE benchmarks with fewer parameters and near machine precision residuals on the inviscid Burgers blowup using Adam.
Regularized Newton's method for neural networks converges exponentially to zero loss with uniform spectral rates in the infinite-width limit via a derived Newton neural tangent kernel.
Deep learning identifies co-triangle-free graphs as e-positive and proves e-positivity for claw-free claw-contractible-free graphs on 10 and 11 vertices, resolving an open conjecture.
AlphaEvolve rediscovered best-known solutions for most of 67 tested math problems and found improved solutions in several cases using LLM-guided evolutionary search.
PINNs with specialized techniques solve the nonlinear Hamiltonian constraint for generic binary black hole initial data, matching traditional NR accuracy.
PINNACLE is an open-source framework for classical and quantum PINNs that supplies modular training methods and benchmarks showing high sensitivity to architecture choices plus parameter-efficiency gains in some hybrid quantum regimes.
G-invariant divergence-free initial data on compact cohomogeneity-one manifolds yield global smooth G-invariant solutions to the incompressible Euler equations.
Hybrid quantum-classical physics-informed neural networks reach accurate solutions to nonlinear PDEs in substantially fewer training epochs than purely classical networks, with larger gains on complex problems.
citing papers explorer
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Neural-Spectral Discovery of Rotating Black Holes Beyond General Relativity
A hybrid neural-spectral method constructs the first fully non-linear rotating black hole solutions in cubic Lovelock gravity, parametric in the gravitational coupling constants.
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Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity II: 3D Profiles, Blowup, and Limiting behavior
Constructs C^α self-similar blowup profiles for 3D Euler vorticity without swirl and proves asymptotically self-similar blowup from C_c^α data, with limiting factorization as α→(1/3)^-.
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Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networks
ASYS recovers known analytical PDE forms and generates new interpretable symbolic approximations, such as a geometric interface formula for 2D Allen-Cahn and a nine-parameter contraction law for Keller-Segel blow-up, via agent-guided evolutionary search on differentiable programs.
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Minimal surfaces, Knots, and Neural Networks
PINNs approximate near-minimal surfaces bounding knots in S^3; their self-intersection numbers align with Fine's conjecture predictions derived from the HOMFLY polynomial.
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Self-similar blow-up solutions of $d$-dimensional incompressible Euler equations with $C^{1,\left(1-2/d\right)-}$ velocity
For every d≥3 and every α<1−2/d, axisymmetric swirl-free incompressible Euler admits self-similar blow-up solutions with C^{1,α} initial velocity that is smooth away from the origin.
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Silent Failures in Physics-Informed Neural Networks: Parameter Poisoning and the Limits of Loss-Based Validation
PINNs trained with incorrect PDE parameters achieve losses matching or beating the correct baseline while their solutions differ by up to 128%.
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On the non-uniqueness of solutions of the axi-symmetric swirl-free Navier-Stokes equations, I
Numerical construction of unstable self-similar axially symmetric swirl-free solutions to the incompressible Navier-Stokes equations on R^3 with global pointwise residuals of order 10^{-10}.
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Fourier Feature Pyramids for Physics-Informed Neural Networks
beignet replaces random Fourier feature embeddings in PINNs with a trainable multi-resolution Fourier feature pyramid, achieving higher accuracy on PDE benchmarks with fewer parameters and near machine precision residuals on the inviscid Burgers blowup using Adam.
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Convergence Analysis of Newton's Method for Neural Networks in the Overparameterized Limit
Regularized Newton's method for neural networks converges exponentially to zero loss with uniform spectral rates in the infinite-width limit via a derived Newton neural tangent kernel.
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How to Use Deep Learning to Identify Sufficient Conditions: A Case Study on Stanley's $e$-Positivity
Deep learning identifies co-triangle-free graphs as e-positive and proves e-positivity for claw-free claw-contractible-free graphs on 10 and 11 vertices, resolving an open conjecture.
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Mathematical exploration and discovery at scale
AlphaEvolve rediscovered best-known solutions for most of 67 tested math problems and found improved solutions in several cases using LLM-guided evolutionary search.
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Solving Hamiltonian Constraint Equation with Physics-Informed Neural Networks
PINNs with specialized techniques solve the nonlinear Hamiltonian constraint for generic binary black hole initial data, matching traditional NR accuracy.
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PINNACLE: An Open-Source Computational Framework for Classical and Quantum PINNs
PINNACLE is an open-source framework for classical and quantum PINNs that supplies modular training methods and benchmarks showing high sensitivity to architecture choices plus parameter-efficiency gains in some hybrid quantum regimes.
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Incompressible Euler fluids on compact cohomogeneity one manifolds
G-invariant divergence-free initial data on compact cohomogeneity-one manifolds yield global smooth G-invariant solutions to the incompressible Euler equations.
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Quantum-Enhanced Convergence of Physics-Informed Neural Networks
Hybrid quantum-classical physics-informed neural networks reach accurate solutions to nonlinear PDEs in substantially fewer training epochs than purely classical networks, with larger gains on complex problems.