Mathematics

73 items

SIAM Journal on Scientific Computing Sep 25, 2026
The authors develop and evaluate a data-free, physics-informed method for learning solution operators: after a finite element discretization, a multilayer perceptron (MLP) maps problem data parameters (boundary conditions, coefficients, right-hand sides, etc.) to the finite element degrees of freedom, with a loss function given by the expected energy functional, plus a parallelizable training algorithm that can use random batches of mesh elements; on a parametric Poisson problem, a Gaussian random field coefficient problem, and a nonlinear neo-Hookean elastic beam, learning errors are small (e.g., for the parametric problem with the full mesh and a 4·512 network, mean relative energy error 2.0e-4, L2 0.0020, H1 0.
The authors develop and evaluate a data-free, physics-informed method for learning solution operators: after a finite element discretization, a multilayer perceptron (MLP) maps problem data parameters (boundary conditions, coefficients, right-hand sides, etc.) to the finite element degrees of freedom, with a loss function given by the expected energy functional, plus a parallelizable training algorithm that can use random batches of mesh elements; on a parametric Poisson problem, a Gaussian random field coefficient problem, and a nonlinear neo-Hookean elastic beam, learning errors are small (e.g., for the parametric problem with the full mesh and a 4·512 network, mean relative energy error 2.0e-4, L2 0.0020, H1 0.
Learning nonlinear finite element solution operators with MLPs and energy minimization yields small learning errors on parametric Poisson, Gaussian random field, and nonlinear elasticity cases, and speeds up Newton's method when the network prediction is used as an initial guess
The authors develop and evaluate a data-free, physics-informed method for learning solution operators: after a finite element discretization, a multilayer perceptron (MLP) maps problem data parameters (boundary conditions, coefficients, right-hand sides, etc.) to the finite element degrees of freedom, with a loss function given by the expected energy functional, plus a parallelizable training algorithm that can use random batches of mesh elements; on a parametric Poisson problem, a Gaussian random field coefficient problem, and a nonlinear neo-Hookean elastic beam, learning errors are small (e.g., for the parametric problem with the full mesh and a 4·512 network, mean relative energy error 2.0e-4, L2 0.0020, H1 0.
The authors develop and evaluate a data-free, physics-informed method for learning solution operators: after a finite element discretization, a multilayer perceptron (MLP) maps problem data parameters (boundary conditions, coefficients, right-hand sides, etc.) to the finite element degrees of freedom, with a loss function given by the expected energy functional, plus a parallelizable training algorithm that can use random batches of mesh elements; on a parametric Poisson problem, a Gaussian random field coefficient problem, and a nonlinear neo-Hookean elastic beam, learning errors are small (e.g., for the parametric problem with the full mesh and a 4·512 network, mean relative energy error 2.0e-4, L2 0.0020, H1 0.
SuperIntelligence - Robotics - Safety & Alignment Sep 25, 2026
This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.
This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.
AI4Math review: from guiding conjectures to formal proofs, AI now solves PKU graduate exams and verifies Erdős counterexamples in Lean
This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.
This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.
The Annals of Applied Probability Sep 25, 2026
The work studies conditional distributions of two Banach-space-valued jointly Gaussian random variables, showing they remain Gaussian and can be determined by a finite-dimensional approximation scheme based on filtering sequences: conditional means converge in the E-norm, covariance operators converge in nuclear norm, conditional probabilities converge weakly, and for continuous Gaussian processes conditioned on partial infinite path observations the mean and covariance functions converge uniformly.
The work studies conditional distributions of two Banach-space-valued jointly Gaussian random variables, showing they remain Gaussian and can be determined by a finite-dimensional approximation scheme based on filtering sequences: conditional means converge in the E-norm, covariance operators converge in nuclear norm, conditional probabilities converge weakly, and for continuous Gaussian processes conditioned on partial infinite path observations the mean and covariance functions converge uniformly.
Steinwart proves with a Banach-space-valued martingale method that conditional distributions of jointly Gaussian variables stay Gaussian and are approximated by finite-dimensional filtering sequences
The work studies conditional distributions of two Banach-space-valued jointly Gaussian random variables, showing they remain Gaussian and can be determined by a finite-dimensional approximation scheme based on filtering sequences: conditional means converge in the E-norm, covariance operators converge in nuclear norm, conditional probabilities converge weakly, and for continuous Gaussian processes conditioned on partial infinite path observations the mean and covariance functions converge uniformly.
The work studies conditional distributions of two Banach-space-valued jointly Gaussian random variables, showing they remain Gaussian and can be determined by a finite-dimensional approximation scheme based on filtering sequences: conditional means converge in the E-norm, covariance operators converge in nuclear norm, conditional probabilities converge weakly, and for continuous Gaussian processes conditioned on partial infinite path observations the mean and covariance functions converge uniformly.
The Annals of Applied Probability Sep 25, 2026
The paper studies multilinear Gibbs measures whose Hamiltonian is a generalized U-statistic with a general base measure; under cut-norm convergence of the coupling matrices it expresses the limiting free energy as an infinite-dimensional optimization over functions, gives sufficient conditions for replica symmetry (constant optimizers) and uses counterexamples to show their necessity, and derives weak limits for local fields, the Hamiltonian and global magnetization, a universal weak law for contrasts n^{-1}Σc_iX_i→0 when Σc_i=o(n), exponential concentration bounds for local and global magnetizations, and existence of a sharp phase transition in the temperature parameter for higher-order interactions.
The paper studies multilinear Gibbs measures whose Hamiltonian is a generalized U-statistic with a general base measure; under cut-norm convergence of the coupling matrices it expresses the limiting free energy as an infinite-dimensional optimization over functions, gives sufficient conditions for replica symmetry (constant optimizers) and uses counterexamples to show their necessity, and derives weak limits for local fields, the Hamiltonian and global magnetization, a universal weak law for contrasts n^{-1}Σc_iX_i→0 when Σc_i=o(n), exponential concentration bounds for local and global magnetizations, and existence of a sharp phase transition in the temperature parameter for higher-order interactions.
Bhattacharya, Deb and Mukherjee write the free-energy limit of multilinear Gibbs measures as an infinite-dimensional optimization, with sufficient conditions and counterexamples for replica symmetry
The paper studies multilinear Gibbs measures whose Hamiltonian is a generalized U-statistic with a general base measure; under cut-norm convergence of the coupling matrices it expresses the limiting free energy as an infinite-dimensional optimization over functions, gives sufficient conditions for replica symmetry (constant optimizers) and uses counterexamples to show their necessity, and derives weak limits for local fields, the Hamiltonian and global magnetization, a universal weak law for contrasts n^{-1}Σc_iX_i→0 when Σc_i=o(n), exponential concentration bounds for local and global magnetizations, and existence of a sharp phase transition in the temperature parameter for higher-order interactions.
The paper studies multilinear Gibbs measures whose Hamiltonian is a generalized U-statistic with a general base measure; under cut-norm convergence of the coupling matrices it expresses the limiting free energy as an infinite-dimensional optimization over functions, gives sufficient conditions for replica symmetry (constant optimizers) and uses counterexamples to show their necessity, and derives weak limits for local fields, the Hamiltonian and global magnetization, a universal weak law for contrasts n^{-1}Σc_iX_i→0 when Σc_i=o(n), exponential concentration bounds for local and global magnetizations, and existence of a sharp phase transition in the temperature parameter for higher-order interactions.
Journal of Global Optimization Sep 25, 2026
This paper studies the generalized trace ratio problem (GTRP), which maximizes a trace-form quadratic fractional objective over the Stiefel manifold, and, using a newly established matrix S-lemma, proves that adding the redundant constraint XX^T⪯I_n together with a well-chosen scaling yields an equivalent problem (GRS) with zero Lagrangian duality gap, whereas the original (GTRP), the redundant-constraint-only version (GR), and the scaling-only version (GS) can all exhibit a positive Lagrangian duality gap.
This paper studies the generalized trace ratio problem (GTRP), which maximizes a trace-form quadratic fractional objective over the Stiefel manifold, and, using a newly established matrix S-lemma, proves that adding the redundant constraint XX^T⪯I_n together with a well-chosen scaling yields an equivalent problem (GRS) with zero Lagrangian duality gap, whereas the original (GTRP), the redundant-constraint-only version (GR), and the scaling-only version (GS) can all exhibit a positive Lagrangian duality gap.
Yang and Xia prove the generalized trace ratio problem needs both a redundant constraint and scaling to close its Lagrangian duality gap
This paper studies the generalized trace ratio problem (GTRP), which maximizes a trace-form quadratic fractional objective over the Stiefel manifold, and, using a newly established matrix S-lemma, proves that adding the redundant constraint XX^T⪯I_n together with a well-chosen scaling yields an equivalent problem (GRS) with zero Lagrangian duality gap, whereas the original (GTRP), the redundant-constraint-only version (GR), and the scaling-only version (GS) can all exhibit a positive Lagrangian duality gap.
This paper studies the generalized trace ratio problem (GTRP), which maximizes a trace-form quadratic fractional objective over the Stiefel manifold, and, using a newly established matrix S-lemma, proves that adding the redundant constraint XX^T⪯I_n together with a well-chosen scaling yields an equivalent problem (GRS) with zero Lagrangian duality gap, whereas the original (GTRP), the redundant-constraint-only version (GR), and the scaling-only version (GS) can all exhibit a positive Lagrangian duality gap.
SIAM Journal on Scientific Computing Sep 25, 2026
This work systematically derives a family of algorithms for the LTLT (L unit lower triangular, T skew-symmetric tridiagonal) triangular tridiagonalization of a skew-symmetric matrix X using the FLAME methodology, presents unpivoted and pivoted blocked right-looking, left-looking, and fused variants, identifies new level-2 and level-3 BLAS-like operations, and implements them with BLIS 2.0 packing mechanisms and OpenMP parallelism; experiments show the best implementations greatly exceed the performance of the only known prior software, PFAPACK and Pfaffine, while matching or exceeding related symmetric factorization software.
This work systematically derives a family of algorithms for the LTLT (L unit lower triangular, T skew-symmetric tridiagonal) triangular tridiagonalization of a skew-symmetric matrix X using the FLAME methodology, presents unpivoted and pivoted blocked right-looking, left-looking, and fused variants, identifies new level-2 and level-3 BLAS-like operations, and implements them with BLIS 2.0 packing mechanisms and OpenMP parallelism; experiments show the best implementations greatly exceed the performance of the only known prior software, PFAPACK and Pfaffine, while matching or exceeding related symmetric factorization software.
FLAME-derived LTLT factorization algorithms for skew-symmetric matrices, with fused BLAS-like operations, greatly outperform PFAPACK and Pfaffine
This work systematically derives a family of algorithms for the LTLT (L unit lower triangular, T skew-symmetric tridiagonal) triangular tridiagonalization of a skew-symmetric matrix X using the FLAME methodology, presents unpivoted and pivoted blocked right-looking, left-looking, and fused variants, identifies new level-2 and level-3 BLAS-like operations, and implements them with BLIS 2.0 packing mechanisms and OpenMP parallelism; experiments show the best implementations greatly exceed the performance of the only known prior software, PFAPACK and Pfaffine, while matching or exceeding related symmetric factorization software.
This work systematically derives a family of algorithms for the LTLT (L unit lower triangular, T skew-symmetric tridiagonal) triangular tridiagonalization of a skew-symmetric matrix X using the FLAME methodology, presents unpivoted and pivoted blocked right-looking, left-looking, and fused variants, identifies new level-2 and level-3 BLAS-like operations, and implements them with BLIS 2.0 packing mechanisms and OpenMP parallelism; experiments show the best implementations greatly exceed the performance of the only known prior software, PFAPACK and Pfaffine, while matching or exceeding related symmetric factorization software.
Terence Tao blog RSS Sep 24, 2026
Using OpenAI's September 8 announcement of a finite-time blow-up proof for the forced Navier-Stokes equation as an entry point, Tapio Schneider distinguishes episteme (explanatory understanding) from techne (the craft of prediction), and argues that when predictions must be trusted before they can be empirically verified—as in decadal climate projection or the design of a novel aircraft—trust comes from an auditable causal chain running from assumptions and input data to outcomes, each link of which can be tested individually; AI should therefore be embedded in auditable scaffolds such as physical conservation laws and used to learn closure models that can be checked against high-resolution simulations, observations, or experiments, rather than deployed as end-to-end models.
Using OpenAI's September 8 announcement of a finite-time blow-up proof for the forced Navier-Stokes equation as an entry point, Tapio Schneider distinguishes episteme (explanatory understanding) from techne (the craft of prediction), and argues that when predictions must be trusted before they can be empirically verified—as in decadal climate projection or the design of a novel aircraft—trust comes from an auditable causal chain running from assumptions and input data to outcomes, each link of which can be tested individually; AI should therefore be embedded in auditable scaffolds such as physical conservation laws and used to learn closure models that can be checked against high-resolution simulations, observations, or experiments, rather than deployed as end-to-end models.
Schneider uses the Navier-Stokes finite-time blow-up proof to argue that AI predictions can be trusted only inside an auditable causal chain
Using OpenAI's September 8 announcement of a finite-time blow-up proof for the forced Navier-Stokes equation as an entry point, Tapio Schneider distinguishes episteme (explanatory understanding) from techne (the craft of prediction), and argues that when predictions must be trusted before they can be empirically verified—as in decadal climate projection or the design of a novel aircraft—trust comes from an auditable causal chain running from assumptions and input data to outcomes, each link of which can be tested individually; AI should therefore be embedded in auditable scaffolds such as physical conservation laws and used to learn closure models that can be checked against high-resolution simulations, observations, or experiments, rather than deployed as end-to-end models.
Using OpenAI's September 8 announcement of a finite-time blow-up proof for the forced Navier-Stokes equation as an entry point, Tapio Schneider distinguishes episteme (explanatory understanding) from techne (the craft of prediction), and argues that when predictions must be trusted before they can be empirically verified—as in decadal climate projection or the design of a novel aircraft—trust comes from an auditable causal chain running from assumptions and input data to outcomes, each link of which can be tested individually; AI should therefore be embedded in auditable scaffolds such as physical conservation laws and used to learn closure models that can be checked against high-resolution simulations, observations, or experiments, rather than deployed as end-to-end models.
Advances in Computational Mathematics Sep 24, 2026
This work studies the Lp error of approximating higher-order Korobov functions f∈K^{m+1}_p(Ω) by deep ReLU convolutional neural networks (CNNs), proving that for depth L≤Csd^4m^3N(log_2 N) there exists a network with inf‖f−f_L‖_{Lp(Ω)}≤C_{m,d}‖D^{m+1}f‖_{Lp(Ω)}N^{−m−1}(log_2 N)^{(m+2)(d−1)}, i.e. it improves the classical second-order mixed-derivative rate O(L^{−2+1/p}) to order (m+1) up to a logarithmic factor, and concludes that the higher-order expressivity of CNNs does not severely suffer from the curse of dimensionality.
This work studies the Lp error of approximating higher-order Korobov functions f∈K^{m+1}_p(Ω) by deep ReLU convolutional neural networks (CNNs), proving that for depth L≤Csd^4m^3N(log_2 N) there exists a network with inf‖f−f_L‖_{Lp(Ω)}≤C_{m,d}‖D^{m+1}f‖_{Lp(Ω)}N^{−m−1}(log_2 N)^{(m+2)(d−1)}, i.e. it improves the classical second-order mixed-derivative rate O(L^{−2+1/p}) to order (m+1) up to a logarithmic factor, and concludes that the higher-order expressivity of CNNs does not severely suffer from the curse of dimensionality.
ReLU CNNs in Korobov spaces lift the approximation order from second order to order m+1, with far weaker dimensional growth than the Sobolev case
This work studies the Lp error of approximating higher-order Korobov functions f∈K^{m+1}_p(Ω) by deep ReLU convolutional neural networks (CNNs), proving that for depth L≤Csd^4m^3N(log_2 N) there exists a network with inf‖f−f_L‖_{Lp(Ω)}≤C_{m,d}‖D^{m+1}f‖_{Lp(Ω)}N^{−m−1}(log_2 N)^{(m+2)(d−1)}, i.e. it improves the classical second-order mixed-derivative rate O(L^{−2+1/p}) to order (m+1) up to a logarithmic factor, and concludes that the higher-order expressivity of CNNs does not severely suffer from the curse of dimensionality.
This work studies the Lp error of approximating higher-order Korobov functions f∈K^{m+1}_p(Ω) by deep ReLU convolutional neural networks (CNNs), proving that for depth L≤Csd^4m^3N(log_2 N) there exists a network with inf‖f−f_L‖_{Lp(Ω)}≤C_{m,d}‖D^{m+1}f‖_{Lp(Ω)}N^{−m−1}(log_2 N)^{(m+2)(d−1)}, i.e. it improves the classical second-order mixed-derivative rate O(L^{−2+1/p}) to order (m+1) up to a logarithmic factor, and concludes that the higher-order expressivity of CNNs does not severely suffer from the curse of dimensionality.
Quanta Magazine Sep 23, 2026
Chemist Gregory Scholes and colleagues, across several papers over the past three years, propose and demonstrate that complex networks of many interacting classical oscillators can give rise to emergent states mathematically describable as vectors in a Hilbert space, thereby mimicking qubits, superposition and interference in a "quantumlike" way, offering an alternative route for quantum biology that does not rely on genuine quantum coherence.
Chemist Gregory Scholes and colleagues, across several papers over the past three years, propose and demonstrate that complex networks of many interacting classical oscillators can give rise to emergent states mathematically describable as vectors in a Hilbert space, thereby mimicking qubits, superposition and interference in a "quantumlike" way, offering an alternative route for quantum biology that does not rely on genuine quantum coherence.
Scholes proposes life may not use real quantum effects but classical oscillating networks that mathematically mimic quantum behavior
Chemist Gregory Scholes and colleagues, across several papers over the past three years, propose and demonstrate that complex networks of many interacting classical oscillators can give rise to emergent states mathematically describable as vectors in a Hilbert space, thereby mimicking qubits, superposition and interference in a "quantumlike" way, offering an alternative route for quantum biology that does not rely on genuine quantum coherence.
Chemist Gregory Scholes and colleagues, across several papers over the past three years, propose and demonstrate that complex networks of many interacting classical oscillators can give rise to emergent states mathematically describable as vectors in a Hilbert space, thereby mimicking qubits, superposition and interference in a "quantumlike" way, offering an alternative route for quantum biology that does not rely on genuine quantum coherence.
arXiv Sep 23, 2026
Lean Pool is a repository of formalized mathematics grown, maintained, and optimized by AI agents; the paper reports that as of September 21, 2026 it holds 211 completed projects and 3,228,485 lines of Lean code from 18 commit contributors, and that agent-assisted dependency upgrades, proof compression, and mathematical review keep these independently developed formalizations usable as Lean and Mathlib evolve.
Lean Pool is a repository of formalized mathematics grown, maintained, and optimized by AI agents; the paper reports that as of September 21, 2026 it holds 211 completed projects and 3,228,485 lines of Lean code from 18 commit contributors, and that agent-assisted dependency upgrades, proof compression, and mathematical review keep these independently developed formalizations usable as Lean and Mathlib evolve.
Lean Pool: An AI-Maintained Archive of Formalized Mathematics
Lean Pool is a repository of formalized mathematics grown, maintained, and optimized by AI agents; the paper reports that as of September 21, 2026 it holds 211 completed projects and 3,228,485 lines of Lean code from 18 commit contributors, and that agent-assisted dependency upgrades, proof compression, and mathematical review keep these independently developed formalizations usable as Lean and Mathlib evolve.
Lean Pool is a repository of formalized mathematics grown, maintained, and optimized by AI agents; the paper reports that as of September 21, 2026 it holds 211 completed projects and 3,228,485 lines of Lean code from 18 commit contributors, and that agent-assisted dependency upgrades, proof compression, and mathematical review keep these independently developed formalizations usable as Lean and Mathlib evolve.

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