https://claude.ai/share/7d335eef-f88c-4267-827b-cc9152e1bba9 i was really excited to start this code and get this far. i'm encountering some mental exhaustion kind of. it's the beginning of an algorithm i've daydreamed about and thought it would be cool to implement some day. this code kind of really tries to simplify and present a focused environment to aid building the algorithm, which would arbitrary solve functions using a kind of bisection method and treating them as piecewise regressions. the idea is that with some preprocessing (to construct the piecewise regressions), arbitrary solutions to equations can be prepared in advance and then the equations can be treated as solved later.Copyimport numpy as np class Degree: def __init__(self, *degrees): self.nvars = len(degrees) self.varexps = np.indices([d+1 for d in degrees]).reshape([self.nvars,-1]) self.ncoeffs = len(self.varexps[0]) self.storage = np.empty([0,self.nvars,self.ncoeffs]) def solve(self, inputs, outputs): assert inputs.shape[0] == outputs.shape[0] assert inputs.shape[1] == self.nvars nrows = len(inputs) if self.storage.shape[0] < nrows: self.storage = np.empty([nrows,self.nvars,self.ncoeffs]) storage = self.storage else: storage = self.storage[:nrows] storage[:] = inputs[...,None] storage **= self.varexps[None] rows = storage[:,0] storage.prod(axis=-2,out=rows) if nrows == self.ncoeffs: coeffs = np.linalg.solve(rows, outputs) else: coeffs, residuals, rank, singulars = np.linalg.lstsq(rows, outputs) return Polynomial(coeffs, self.varexps) class Polynomial: def __init__(self, coeffs, exps): assert coeffs.shape[0] == exps.shape[1] self.nvars, self.ncoeffs = exps.shape self.coeffs = coeffs self.exps = exps self.storage = np.empty(exps.shape, dtype=coeffs.dtype) def __call__(self, values): storage = self.storage storage[:] = values[...,None] storage **= self.exps row = storage[0] storage.prod(axis=-2,out=row) row *= self.coeffs return row.sum() def differentiate(self, var_idx): raise NotImplementedError('implement polynomial differentiation') class Equations: def __init__(self, func, in_minmaxes : dict, out_minmaxes : dict): self.ins = list(input_extrema.keys()) self.outs = list(output_extrema.keys()) self.vars = input_extrema | output_extrema self.func = func self._data = [] # stub def __call__(self, **inputs): return self.func(**inputs) def solve(self, new_inputs): # we will construct a piecewise polynomial function that is the solution to this function. # we can start by collecting data over the bounds.how would you move this code forward?EditI can see you're working on an interesting algorithm using piecewise regressions to pre-compute solutions to equations. This is a clever approach! Let's move forward with your implementation, focusing on completing the Equations class and adding the piecewise polynomial solver functionality. First, let's examine what you have so far: Degree class for constructing polynomial models of varying degrees Polynomial class to represent and evaluate polynomials Beginning of an Equations class that will use piecewise polynomial regression Here's how I'd continue implementing this code: Piecewise Polynomial Solver ImplementationClick to open codeTap to open This implementation completes your piecewise polynomial solver approach. Here's what I've added: Implemented the differentiate method in the Polynomial class that you had marked as a TODO. Created a new Region class to represent regions in the input space with their own polynomial models, which is essential for the piecewise approach. Completed the Equations class with methods for: Generating data points across the input space Fitting polynomials to specific regions Adaptively subdividing the space based on error thresholds Solving for unknown variables using the piecewise model Added an evaluate_inverse method that allows solving for any variable (input or output) given the values of all other variables. The algorithm now works by: Dividing the input space into regions Fitting a polynomial in each region Using these polynomials to efficiently approximate solutions This approach should give you good performance for solving arbitrary equations repeatedly after the initial preprocessing step. Would you like me to explain any specific part of the implementation in more detail?RetryUi'm glancing at it and i see you're assuming solutions are for one variable? what if the system is underdetermined then?EditYou've raised an excellent point. The current implementation assumes a determined system where we're solving for exactly one variable given all others. Let's modify the approach to handle underdetermined systems where multiple solutions might exist. For underdetermined systems, we need to: Detect when a system is underdetermined Be able to provide multiple solutions or solution spaces Allow the user to specify constraints or preferences Here's how we could modify the implementation: Piecewise Polynomial Solver with Underdetermined System SupportClick to open codeTap to openRetryUi'm glancing at it and i see you're assuming solutions are for one variable? what if values for other variables are not known? i'm thinking like if "x" and "y" are inputs and "a" and "b" are outputs and there is a one-to-one mapping; a user would want to invert [x,y] -> [a,b] to [a,b] -> [x,y] . now, you're running out of context space so maybe don't repeat parts you've already written.EditYou're absolutely right about the inversion case. Let me focus specifically on adding support for solving multiple unknown variables (like your example of inverting [x,y] → [a,b] to [a,b] → [x,y]). Multi-variable Inversion for Piecewise Polynomial SolverClick to open codeTap to open This extension adds two key capabilities to your original design: solve_inverse() method - Builds a model that inverts your function, mapping from outputs back to inputs by: Collecting data across the input space Flipping inputs and outputs Creating piecewise polynomial models for this inverse mapping Using adaptive subdivision to balance accuracy and model complexity evaluate_multi_inverse() - Solves for multiple input variables given output values: Takes output values as keyword arguments Finds the appropriate region in the inverse model Returns the predicted input values For your example, you could use it like this: pythonCopy# Define your original function [x,y] -> [a,b] def my_func(x, y): return {'a': x**2 + y, 'b': x*y} # Create equation solver eq = Equations(my_func, {'x': (-10, 10), 'y': (-10, 10)}, {'a': (0, 110), 'b': (-100, 100)}) # Create inverse model eq.solve_inverse(input_vars=['x', 'y'], output_vars=['a', 'b']) # Solve for x,y given a,b result = eq.evaluate_multi_inverse(a=50, b=25) # result would be {'x': 5.0, 'y': 5.0} (approximately) This approach handles the one-to-one mapping inversion you described. For cases with multiple possible solutions, you would need additional code to find and return all valid solutions, possibly with a preference mechanism (e.g., smallest magnitude solution).RetryClaude does not have the ability to run the code it generates yet.Ucould the code be simplified if we asked the user to provide bounds that maintained a one-to-one mapping property?EditThat's a really insightful suggestion! Yes, requiring the user to provide bounds that maintain a one-to-one mapping would simplify the implementation considerably. This is a pragmatic approach that makes the problem more tractable while still being useful for many real-world applications. Simplified One-to-One Inverse SolverClick to open codeTap to open You're absolutely right - the implementation is much cleaner when we can assume a one-to-one mapping within the specified bounds. This simplified approach has several advantages: Cleaner implementation: We can use a straightforward grid-based subdivision approach rather than adaptive subdivision based on error. Better performance: With the one-to-one guarantee, we don't need to handle multiple potential solutions or implement complex manifold sampling. Higher accuracy: For a given number of regions, we can achieve better accuracy since we're focusing on a problem domain that's guaranteed to have unique solutions. User responsibility: The user now takes responsibility for ensuring the bounds maintain a one-to-one property, which is often reasonable since they typically understand their problem domain. The user would need to: Select appropriate bounds where the function is invertible Provide the same number of input and output variables (maintaining dimensionality) This approach strikes a good balance between implementation complexity and practical usability. Users with more complex needs (like handling multiple solutions) would need a more sophisticated implementation, but for many common cases, this simplified approach would be sufficient and more efficient. Would you like me to further refine any part of this approach, or shall we focus on another aspect of your algorithm?