我正在使用tfp.math.ode.BDF对常微分方程组(ODEs(进行数值积分。在这里看我的合作笔记本。
与API文档中的示例代码一样，函数ode_fn(t, y, theta)定义了要解决的ODE系统。我能够取ode_fn与theta的梯度，并将ODE与tfp.math.ode.BDF积分。
然而，当我尝试将ODE解决方案的梯度结果写入theta时，我会得到以下错误。当我用一组更简单的ODE替换ode_fn时，代码运行起来没有任何问题。是否应调整解算器设置以避免此错误？
InvalidArgumentError Traceback (most recent call last)
<ipython-input-9-77ebcb7dd888> in <module>()
----> 1 print(g.gradient(foo, theta0))
5 frames
/usr/local/lib/python3.7/dist-packages/tensorflow/python/eager/backprop.py in gradient(self, target, sources, output_gradients, unconnected_gradients)
1088 output_gradients=output_gradients,
1089 sources_raw=flat_sources_raw,
-> 1090 unconnected_gradients=unconnected_gradients)
1091
1092 if not self._persistent:
/usr/local/lib/python3.7/dist-packages/tensorflow/python/eager/imperative_grad.py in imperative_grad(tape, target, sources, output_gradients, sources_raw, unconnected_gradients)
75 output_gradients,
76 sources_raw,
---> 77 compat.as_str(unconnected_gradients.value))
/usr/local/lib/python3.7/dist-packages/tensorflow/python/eager/function.py in _backward_function_wrapper(*args)
1301 break
1302 return backward._call_flat( # pylint: disable=protected-access
-> 1303 processed_args, remapped_captures)
1304
1305 return _backward_function_wrapper, recorded_outputs
/usr/local/lib/python3.7/dist-packages/tensorflow/python/eager/function.py in _call_flat(self, args, captured_inputs, cancellation_manager)
1962 # No tape is watching; skip to running the function.
1963 return self._build_call_outputs(self._inference_function.call(
-> 1964 ctx, args, cancellation_manager=cancellation_manager))
1965 forward_backward = self._select_forward_and_backward_functions(
1966 args,
/usr/local/lib/python3.7/dist-packages/tensorflow/python/eager/function.py in call(self, ctx, args, cancellation_manager)
594 inputs=args,
595 attrs=attrs,
--> 596 ctx=ctx)
597 else:
598 outputs = execute.execute_with_cancellation(
/usr/local/lib/python3.7/dist-packages/tensorflow/python/eager/execute.py in quick_execute(op_name, num_outputs, inputs, attrs, ctx, name)
58 ctx.ensure_initialized()
59 tensors = pywrap_tfe.TFE_Py_Execute(ctx._handle, device_name, op_name,
---> 60 inputs, attrs, num_outputs)
61 except core._NotOkStatusException as e:
62 if name is not None:
InvalidArgumentError: Input matrix is not invertible.
[[{{node gradients/IdentityN_grad/bdfGradients/while/body/_718/gradients/IdentityN_grad/bdfGradients/while/bdf/while/body/_2247/gradients/IdentityN_grad/bdfGradients/while/bdf/while/while/body/_3245/gradients/IdentityN_grad/bdfGradients/while/bdf/while/while/while/body/_5670/gradients/IdentityN_grad/bdfGradients/while/bdf/while/while/while/while/body/_7588/gradients/IdentityN_grad/bdfGradients/while/bdf/while/while/while/while/triangular_solve/MatrixTriangularSolve}}]] [Op:__inference___backward_debug_ode_solver_9192_32890]
Function call stack:
__backward_debug_ode_solver_9192
我通过以下操作避开了这些数字错误：
- 重新缩放我的因变量和参数，使一切都在几个数量级以内
- 增加ODE解算器返回因变量值的自变量值的数量。我只需要20个均匀间隔的时间点来计算损失函数，但如果我想通过参数计算损失函数的梯度，ODE求解器必须在时间上返回至少60个均匀间隔点。我只是丢掉了额外的分数